Chapter 1. Functions and Their Graphs. Selected Applications

Transcription

1 Chapter Functions and Their Graphs. Lines in the Plane. Functions. Graphs of Functions. Shifting, Reflecting, and Stretching Graphs.5 Combinations of Functions. Inverse Functions.7 Linear Models and Scatter Plots Selected Applications Functions have man real-life applications. The applications listed below represent a small sample of the applications in this chapter. Rental Demand, Eercise 8, page Postal Regulations, Eercise 77, page 7 Motor Vehicles, Eercise 8, page 8 Fluid Flow, Eercise 9, page 0 Finance, Eercise 58, page 50 Bacteria, Eercise 8, page Consumer Awareness, Eercise 8, page Shoe Sizes, Eercises 0 and 0, page 7 Cell Phones, Eercise, page 79 An equation in and defines a relationship between the two variables. The equation ma be represented as a graph, providing another perspective on the relationship between and. In Chapter, ou will learn how to write and graph linear equations, how to evaluate and find the domains and ranges of functions, and how to graph functions and their transformations. Inde Stock Imager Refrigeration slows down the activit of bacteria in food so that it takes longer for the bacteria to spoil the food. The number of bacteria in a refrigerated food is a function of the amount of time the food has been out of refrigeration. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

2 Chapter Functions and Their Graphs Introduction to Librar of Parent Functions In Chapter, ou will be introduced to the concept of a function. As ou proceed through the tet, ou will see that functions pla a primar role in modeling real-life situations. There are three basic tpes of functions that have proven to be the most important in modeling real-life situations. These functions are algebraic functions, eponential and logarithmic functions, and trigonometric and inverse trigonometric functions. These three tpes of functions are referred to as the elementar functions, though the are often placed in the two categories of algebraic functions and transcendental functions. Each time a new tpe of function is studied in detail in this tet, it will be highlighted in a bo similar to this one. The graphs of man of these functions are shown on the inside front cover of this tet. A review of these functions can be found in the Stud Capsules. Algebraic Functions These functions are formed b appling algebraic operations to the identit function f. Name Function Location Linear f a b Section. Quadratic f a b c Section. Cubic f a b c d Section. Polnomial P a n n a n n... a a a 0 Section. Rational f N N and D are polnomial functions D, Section. Radical f P n Section. Transcendental Functions These functions cannot be formed from the identit function b using algebraic operations. Name Function Location Eponential f a, a > 0, a Section. Logarithmic f log a, > 0, a > 0, a Section. Trigonometric f sin, f cos, f tan, f csc, f sec, f cot Section. Inverse Trigonometric f arcsin, f arccos, f arctan Section.7 Nonelementar Functions Some useful nonelementar functions include the following. Name Function Location Absolute value f g, g is an elementar function Section. Piecewise-defined f,, < Section. Greatest integer f g, g is an elementar function Section. Data defined Formula for temperature: F 9 C Section. 5 Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

3 Section. Lines in the Plane. Lines in the Plane The Slope of a Line In this section, ou will stud lines and their equations. The slope of a nonvertical line represents the number of units the line rises or falls verticall for each unit of horizontal change from left to right. For instance, consider the two points and,, on the line shown in Figure.. As ou move from left to right along this line, a change of units in the vertical direction corresponds to a change of units in the horizontal direction. That is, and the change in the change in. The slope of the line is given b the ratio of these two changes. What ou should learn Find the slopes of lines. Write linear equations given points on lines and their slopes. Use slope-intercept forms of linear equations to sketch lines. Use slope to identif parallel and perpendicular lines. Wh ou should learn it The slope of a line can be used to solve real-life problems. For instance, in Eercise 87 on page. ou will use a linear equation to model student enrollment at Penn State Universit. (, ) (, ) Sk Bonillo/PhotoEdit Figure. Definition of the Slope of a Line The slope m of the nonvertical line through and,, is m change in change in where. When this formula for slope is used, the order of subtraction is important. Given two points on a line, ou are free to label either one of them as, and the other as,. However, once ou have done this, ou must form the numerator and denominator using the same order of subtraction. m m m Correct Correct Incorrect Throughout this tet, the term line alwas means a straight line. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

4 Chapter Functions and Their Graphs Eample Finding the Slope of a Line Find the slope of the line passing through each pair of points. a., 0 and, b., and, c. 0, and, a. b. c. Difference in -values m 0 5 Difference in -values m 0 0 m The graphs of the three lines are shown in Figure.. Note that the square setting gives the correct steepness of the lines. Eploration Use a graphing utilit to compare the slopes of the lines 0.5,,, and. What do ou observe about these lines? Compare the slopes of the lines 0.5,,, and. What do ou observe about these lines? (Hint: Use a square setting to guarantee a true geometric perspective.) (, ) 5 (, 0) (, ) (, ) 5 (0, ) 8 (, ) (a) (b) (c) Figure. Now tr Eercise 9. The definition of slope does not appl to vertical lines. For instance, consider the points, and, on the vertical line shown in Figure.. Appling the formula for slope, ou obtain m 0. Undefined Because division b zero is undefined, the slope of a vertical line is undefined. From the slopes of the lines shown in Figures. and., ou can make the following generalizations about the slope of a line. 5 (, ) (, ) 8 Figure. The Slope of a Line. A line with positive slope m > 0 rises from left to right.. A line with negative slope m < 0 falls from left to right.. A line with zero slope m 0 is horizontal.. A line with undefined slope is vertical. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

5 Section. Lines in the Plane 5 The Point-Slope Form of the Equation of a Line If ou know the slope of a line and ou also know the coordinates of one point on the line, ou can find an equation for the line. For instance, in Figure., let, be a point on the line whose slope is m. If, is an other point on the line, it follows that (, ) (, ) m. This equation in the variables and can be rewritten in the point-slope form of the equation of a line. Figure. Point-Slope Form of the Equation of a Line The point-slope form of the equation of the line that passes through the point, and has a slope of m is m. The point-slope form is most useful for finding the equation of a line if ou know at least one point that the line passes through and the slope of the line. You should remember this form of the equation of a line. Eample The Point-Slope Form of the Equation of a Line Find an equation of the line that passes through the point, and has a slope of. m Point-slope form Substitute for, m, and. Simplif. 5 Solve for. The line is shown in Figure.5. Now tr Eercise 5. The point-slope form can be used to find an equation of a nonvertical line passing through two points, and,. First, find the slope of the line. m, Then use the point-slope form to obtain the equation. This is sometimes called the two-point form of the equation of a line Figure.5 = 5 (, ) STUDY TIP When ou find an equation of the line that passes through two given points, ou need to substitute the coordinates of onl one of the points into the point-slope form. It does not matter which point ou choose because both points will ield the same result. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

6 Chapter Functions and Their Graphs Eample A Linear Model for Sales Prediction During 00, Nike s net sales were $.5 billion, and in 005 net sales were $.7 billion. Write a linear equation giving the net sales in terms of the ear. Then use the equation to predict the net sales for 00. (Source: Nike, Inc.) Let 0 represent 000. In Figure., let,.5 and 5,.7 be two points on the line representing the net sales. The slope of this line is m Librar of Parent Functions: Linear Function In the net section, ou will be introduced to the precise meaning of the term function. The simplest tpe of function is a linear function of the form f m b. As its name implies, the graph of a linear function is a line that has a slope of m and a -intercept at 0, b. The basic characteristics of a linear function are summarized below. (Note that some of the terms below will be defined later in the tet.) A review of linear functions can be found in the Stud Capsules. Graph of f m b, m > 0 Graph of f m b, m < 0 Domain:, Domain:, Range:, Range:, -intercept: b m, 0 -intercept: b m, 0 -intercept: 0, b -intercept: 0, b Increasing Decreasing f() = m + b, m > 0 ( b, 0 m ( (0, b).9. m B the point-slope form, the equation of the line is as follows..5.9 Write in point-slope form..9.9 Simplif. Now, using this equation, ou can predict the 00 net sales to be $5. billion. Now tr Eercise 5. (0, b) When m 0, the function f b is called a constant function and its graph is a horizontal line. ( f() = m + b, m < 0 b, 0 m ( 0 (, 5.) (,.5) (5,.7) = Figure. STUDY TIP The prediction method illustrated in Eample is called linear etrapolation. Note in the top figure below that an etrapolated point does not lie between the given points. When the estimated point lies between two given points, as shown in the bottom figure, the procedure used to predict the point is called linear interpolation. Linear Etrapolation Given points Given points Linear Interpolation 8 Estimated point Estimated point Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

7 Sketching Graphs of Lines Man problems in coordinate geometr can be classified as follows.. Given a graph (or parts of it), find its equation.. Given an equation, sketch its graph. For lines, the first problem is solved easil b using the point-slope form. This formula, however, is not particularl useful for solving the second tpe of problem. The form that is better suited to graphing linear equations is the slope-intercept form of the equation of a line, m b. Section. Lines in the Plane 7 Slope-Intercept Form of the Equation of a Line The graph of the equation m b is a line whose slope is m and whose -intercept is 0, b. Eample Using the Slope-Intercept Form Determine the slope and -intercept of each linear equation. Then describe its graph. a. b. Algebraic a. Begin b writing the equation in slope-intercept form. Write original equation. Subtract from each side. Write in slope-intercept form. From the slope-intercept form of the equation, the slope is and the -intercept is 0,. Because the slope is negative, ou know that the graph of the equation is a line that falls one unit for ever unit it moves to the right. b. B writing the equation in slope-intercept form 0 ou can see that the slope is 0 and the -intercept is 0,. A zero slope implies that the line is horizontal. Graphical a. Solve the equation for to obtain. Enter this equation in our graphing utilit. Use a decimal viewing window to graph the equation. To find the -intercept, use the value or trace feature. When 0,, as shown in Figure.7(a). So, the -intercept is 0,. To find the slope, continue to use the trace feature. Move the cursor along the line until. At this point,. So the graph falls unit for ever unit it moves to the right, and the slope is. b. Enter the equation in our graphing utilit and graph the equation. Use the trace feature to verif the -intercept 0,, as shown in Figure.7(b), and to see that the value of is the same for all values of. So, the slope of the horizontal line is Now tr Eercise 7. (a) Figure.7. (b) Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

8 8 Chapter Functions and Their Graphs From the slope-intercept form of the equation of a line, ou can see that a horizontal line m 0 has an equation of the form b. This is consistent with the fact that each point on a horizontal line through 0, b has a -coordinate of b. Similarl, each point on a vertical line through a, 0 has an -coordinate of a. So, a vertical line has an equation of the form a. This equation cannot be written in slope-intercept form because the slope of a vertical line is undefined. However, ever line has an equation that can be written in the general form A B C 0 where A and B are not both zero. General form of the equation of a line Eploration Graph the lines,, and in the same viewing window. What do ou observe? Graph the lines,, and in the same viewing window. What do ou observe? Summar of Equations of Lines. General form: A B C 0. Vertical line: a. Horizontal line: b. Slope-intercept form: m b 5. Point-slope form: m Eample 5 Different Viewing Windows The graphs of the two lines and 0 are shown in Figure.8. Even though the slopes of these lines are quite different ( and 0, respectivel), the graphs seem misleadingl similar because the viewing windows are different. = = = (a) = + Figure.8 0 Now tr Eercise 5. 0 (b) 0 TECHNOLOGY TIP When a graphing utilit is used to graph a line, it is important to realize that the graph of the line ma not visuall appear to have the slope indicated b its equation. This occurs because of the viewing window used for the graph. For instance, Figure.9 shows graphs of produced on a graphing utilit using three different viewing windows. Notice that the slopes in Figures.9(a) and (b) do not visuall appear to be equal to. However, if ou use a square setting, as in Figure.9(c), the slope visuall appears to be (c) Figure.9 0 = + Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

9 Section. Lines in the Plane 9 Parallel and Perpendicular Lines The slope of a line is a convenient tool for determining whether two lines are parallel or perpendicular. Parallel Lines Two distinct nonvertical lines are parallel if and onl if their slopes are equal. That is, Eample m m. Equations of Parallel Lines Find the slope-intercept form of the equation of the line that passes through the point, and is parallel to the line 5. Begin b writing the equation of the given line in slope-intercept form Write original equation. Multipl b. Add to each side. TECHNOLOGY TIP Be careful when ou graph equations such as 7 with our graphing utilit. A common mistake is to tpe in the equation as Y X 7 which ma not be interpreted b our graphing utilit as the original equation. You should use one of the following formulas. Y X 7 Y X 7 Do ou see wh? 5 Write in slope-intercept form. Therefore, the given line has a slope of m. An line parallel to the given line must also have a slope of. So, the line through, has the following equation. Write in point-slope form. = 5 7 Simplif. Write in slope-intercept form. 5 (, ) Notice the similarit between the slope-intercept form of the original equation and the slope-intercept form of the parallel equation. The graphs of both equations are shown in Figure.0. Now tr Eercise 57(a). Figure.0 = 7 Perpendicular Lines Two nonvertical lines are perpendicular if and onl if their slopes are negative reciprocals of each other. That is, m m. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

10 0 Chapter Functions and Their Graphs Eample 7 Equations of Perpendicular Lines Find the slope-intercept form of the equation of the line that passes through the point, and is perpendicular to the line 5. From Eample, ou know that the equation can be written in the slope-intercept form 5. You can see that the line has a slope of So, an line perpendicular to this line must have a slope of because. is the negative reciprocal of. So, the line through the point, has the following equation. Write in point-slope form. Simplif. Write in slope-intercept form. The graphs of both equations are shown in Figure.. Now tr Eercise 57(b). 7 (, ) Figure. = = + 5 Eample 8 Graphs of Perpendicular Lines Use a graphing utilit to graph the lines and in the same viewing window. The lines are supposed to be perpendicular (the have slopes of m and m ). Do the appear to be perpendicular on the displa? If the viewing window is nonsquare, as in Figure., the two lines will not appear perpendicular. If, however, the viewing window is square, as in Figure., the lines will appear perpendicular. = + = + 0 = + = Figure. Figure. Now tr Eercise 7. 0 Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

11 Section. Lines in the Plane. Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check. Match each equation with its form. (a) A B C 0 (b) a (c) b (d) m b (e) m (i) vertical line (ii) slope-intercept form (iii) general form (iv) point-slope form (v) horizontal line In Eercises 5, fill in the blanks.. For a line, the ratio of the change in to the change in is called the of the line.. Two lines are if and onl if their slopes are equal.. Two lines are if and onl if their slopes are negative reciprocals of each other. 5. The prediction method is the method used to estimate a point on a line that does not lie between the given points. In Eercises and, identif the line that has the indicated slope.. (a) m (b) m is undefined. (c) m. (a) m 0 (b) m (c) m Figure for Figure for In Eercises and, sketch the lines through the point with the indicated slopes on the same set of coordinate aes. Point Slopes., (a) 0 (b) (c) (d)., (a) (b) (c) (d) Undefined In Eercises 5 and, estimate the slope of the line L L 8 L 8 L 8 L L In Eercises 7 0, find the slope of the line passing through the pair of points. Then use a graphing utilit to plot the points and use the draw feature to graph the line segment connecting the two points. (Use a square setting.) 7. 0, 0,, 0 8.,,, 9.,,, 0.,,, In Eercises 8, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are man correct answers.) Point Slope...,,, 5 m 0 m 0 m is undefined.., m is undefined , 9 5, 7,, m m m m In Eercises 9, (a) find the slope and -intercept (if possible) of the equation of the line algebraicall, and (b) sketch the line b hand. Use a graphing utilit to verif our answers to parts (a) and (b) Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

12 Chapter Functions and Their Graphs In Eercises 5, find the general form of the equation of the line that passes through the given point and has the indicated slope. Sketch the line b hand. Use a graphing utilit to verif our sketch, if possible. Point Slope ,,,, 5, m m m m m is undefined. 0. 0, m is undefined..,.., 8.5 m 0 m 0 In Eercises, find the slope-intercept form of the equation of the line that passes through the points. Use a graphing utilit to graph the line.. 5,, 5, 5.,,, 5. 8,, 8, 7.,,, 7.,,, 5 8.,,, 9. 0, 5, 9 0, ,,, 7., 0.,, 0.. 8, 0.,,. In Eercises and, find the slope-intercept form of the equation of the line shown... (, ) (, 7), ) 5. Annual Salar A jeweler s salar was $8,500 in 00 and $,900 in 00. The jeweler s salar follows a linear growth pattern. What will the jeweler s salar be in 008?. Annual Salar A librarian s salar was $5,000 in 00 and $7,500 in 00. The librarian s salar follows a linear growth pattern. What will the librarian s salar be in 008? ) (, ) In Eercises 7 50, determine the slope and -intercept of the linear equation. Then describe its graph In Eercises 5 and 5, use a graphing utilit to graph the equation using each of the suggested viewing windows. Describe the difference between the two graphs Xmin = -5 Xma = 0 Xscl = Ymin = - Yma = 0 Yscl = 8 5 Xmin = -5 Xma = 5 Xscl = Ymin = -0 Yma = 0 Yscl = Xmin = - Xma = 0 Xscl = Ymin = - Yma = Yscl = Xmin = -5 Xma = 0 Xscl = Ymin = -80 Yma = 80 Yscl = 0 In Eercises 5 5, determine whether the lines and passing through the pairs of points are parallel, perpendicular, or neither. 5. L : 0,, 5, 9 5. L :,,, 5 L : 0,,, L :,, 5, L :,,, 0 5. L : (, 8), (, ) L : 0,, 5, 7 L :, 5,, In Eercises 57, write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line. Point Line 57., 58., 7 59., ,. 9., 0., 0 L L Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

13 Section. Lines in the Plane In Eercises and, the lines are parallel. Find the slopeintercept form of the equation of line... 5 = + (, ) (, ) = + (b) Find the equation of the line between the ears 995 and 00. (c) Interpret the meaning of the slope of the equation from part (b) in the contet of the problem. (d) Use the equation from part (b) to estimate the earnings per share of stock in the ear 00. Do ou think this is an accurate estimation? Eplain. 7. Sales The graph shows the sales (in billions of dollars) for Goodear Tire for the ears 995 through 00, where t 5 represents 995. (Source: Goodear Tire) In Eercises 5 and, the lines are perpendicular. Find the slope-intercept form of the equation of line. 5.. (, ) 5 = + (, 5) = Sales (in billions of dollars) (7,.) (, 5.) (0,.) (5,.) (,.9) (,.) (9,.9) (,.) (8,.) Year (5 995) (, 8.) Graphical Analsis In Eercises 7 70, identif an relationships that eist among the lines, and then use a graphing utilit to graph the three equations in the same viewing window. Adjust the viewing window so that each slope appears visuall correct. Use the slopes of the lines to verif our results. 7. (a) (b) (c) 8. (a) (b) (c) 9. (a) (b) (c) 70. (a) 8 (b) (c) 7. Earnings per Share The graph shows the earnings per share of stock for Circuit Cit for the ears 995 through 00. (Source: Circuit Cit Stores, Inc.) Earnings per share (in dollars) (5, 0.9) (8, 0.7) (, 0.9) (7, 0.57) (9,.0) (0, 0.8) (, 0.0) Year (5 995) (, 0.9) (, 0.) (, 0.00) (a) Use the slopes to determine the ears in which the earnings per share of stock showed the greatest increase and greatest decrease. (a) Use the slopes to determine the ears in which the sales for Goodear Tire showed the greatest increase and the smallest increase. (b) Find the equation of the line between the ears 995 and 00. (c) Interpret the meaning of the slope of the equation from part (b) in the contet of the problem. (d) Use the equation from part (b) to estimate the sales for Goodear Tire in the ear 00. Do ou think this is an accurate estimation? Eplain. 7. Height The rise to run ratio of the roof of a house determines the steepness of the roof. The rise to run ratio of the roof in the figure is to. Determine the maimum height in the attic of the house if the house is feet wide. ft attic height 7. Road Grade When driving down a mountain road, ou notice warning signs indicating that it is a % grade. This means that the slope of the road is 00. Approimate the amount of horizontal change in our position if ou note from elevation markers that ou have descended 000 feet verticall. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

14 Chapter Functions and Their Graphs Rate of Change In Eercises 75 78, ou are given the dollar value of a product in 00 and the rate at which the value of the product is epected to change during the net 5 ears. Write a linear equation that gives the dollar value V of the product in terms of the ear t. (Let t represent 00.) 00 Value Rate 75. $50 $5 increase per ear 7. $5 $.50 increase per ear 77. $0,00 $000 decrease per ear 78. $5,000 $500 decrease per ear Graphical Interpretation In Eercises 79 8, match the description with its graph. Determine the slope of each graph and how it is interpreted in the given contet. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) You are paing $0 per week to repa a $00 loan. 80. An emploee is paid $.50 per hour plus $.50 for each unit produced per hour. 8. A sales representative receives $0 per da for food plus $.5 for each mile traveled. 8. A computer that was purchased for $00 depreciates $00 per ear. 8. Depreciation A school district purchases a high-volume printer, copier, and scanner for $5,000. After 0 ears, the equipment will have to be replaced. Its value at that time is epected to be $000. (a) Write a linear equation giving the value V of the equipment during the 0 ears it will be used. (b) Use a graphing utilit to graph the linear equation representing the depreciation of the equipment, and use the value or trace feature to complete the table. (b) (d) t V (c) Verif our answers in part (b) algebraicall b using the equation ou found in part (a) Meteorolog Recall that water freezes at 0 C F and boils at 00 C F. (a) Find an equation of the line that shows the relationship between the temperature in degrees Celsius C and degrees Fahrenheit F. (b) Use the result of part (a) to complete the table. C F Cost, Revenue, and Profit A contractor purchases a bulldozer for $,500. The bulldozer requires an average ependiture of $5.5 per hour for fuel and maintenance, and the operator is paid $.50 per hour. (a) Write a linear equation giving the total cost C of operating the bulldozer for t hours. (Include the purchase cost of the bulldozer.) (b) Assuming that customers are charged $7 per hour of bulldozer use, write an equation for the revenue R derived from t hours of use. (c) Use the profit formula P R C to write an equation for the profit derived from t hours of use. (d) Use the result of part (c) to find the break-even point (the number of hours the bulldozer must be used to ield a profit of 0 dollars). 8. Rental Demand A real estate office handles an apartment comple with 50 units. When the rent per unit is $580 per month, all 50 units are occupied. However, when the rent is $5 per month, the average number of occupied units drops to 7. Assume that the relationship between the monthl rent p and the demand is linear. (a) Write the equation of the line giving the demand in terms of the rent p. (b) Use a graphing utilit to graph the demand equation and use the trace feature to estimate the number of units occupied when the rent is $55. Verif our answer algebraicall. (c) Use the demand equation to predict the number of units occupied when the rent is lowered to $595. Verif our answer graphicall. 87. Education In 99, Penn State Universit had an enrollment of 75,9 students. B 005, the enrollment had increased to 80,. (Source: Penn State Fact Book) (a) What was the average annual change in enrollment from 99 to 005? (b) Use the average annual change in enrollment to estimate the enrollments in 98, 997, and 000. (c) Write the equation of a line that represents the given data. What is its slope? Interpret the slope in the contet of the problem. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

15 Section. Lines in the Plane Writing Using the results of Eercise 87, write a short paragraph discussing the concepts of slope and average rate of change. Snthesis True or False? In Eercises 89 and 90, determine whether the statement is true or false. Justif our answer. 89. The line through 8, and, and the line through 0, and 7, 7 are parallel. 90. If the points 0, and, 9 lie on the same line, then the point, 7 also lies on that line. Eploration In Eercises 9 9, use a graphing utilit to graph the equation of the line in the form a, b a 0, b 0. Use the graphs to make a conjecture about what a and b represent. Verif our conjecture In Eercises 95 98, use the results of Eercises 9 9 to write an equation of the line that passes through the points intercept:, intercept: 5, 0 -intercept: 0, -intercept: 0, 97. -intercept:, intercept:, 0 -intercept: 0, -intercept: 0, 5 Librar of Parent Functions In Eercises 99 and 00, determine which equation(s) ma be represented b the graph shown. (There ma be more than one correct answer.) Librar of Parent Functions In Eercises 0 and 0, determine which pair of equations ma be represented b the graphs shown (a) 5 (b) 5 (c) 5 (d) 5 (a) (b) (c) (d) 0. Think About It Does ever line have both an -intercept and a -intercept? Eplain. 0. Think About It Can ever line be written in slope-intercept form? Eplain. 05. Think About It Does ever line have an infinite number of lines that are parallel to the given line? Eplain. 0. Think About It Does ever line have an infinite number of lines that are perpendicular to the given line? Eplain. Skills Review In Eercises 07, determine whether the epression is a polnomial. If it is, write the polnomial in standard form In Eercises, factor the trinomial (a) (b) (c) (d) (a) (b) (c) (d) Make a Decision To work an etended application analzing the numbers of bachelor s degrees earned b women in the United States from 985 to 005, visit this tetbook s Online Stud Center. (Data Source: U.S. Census Bureau) The Make a Decision eercise indicates a multipart eercise using large data sets. Go to this tetbook s Online Stud Center to view these eercises. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

16 Chapter Functions and Their Graphs. Functions Introduction to Functions Man everda phenomena involve pairs of quantities that are related to each other b some rule of correspondence. The mathematical term for such a rule of correspondence is a relation. Here are two eamples.. The simple interest I earned on an investment of $000 for ear is related to the annual interest rate r b the formula I 000r.. The area A of a circle is related to its radius r b the formula A r. Not all relations have simple mathematical formulas. For instance, people commonl match up NFL starting quarterbacks with touchdown passes, and hours of the da with temperature. In each of these cases, there is some relation that matches each item from one set with eactl one item from a different set. Such a relation is called a function. What ou should learn Decide whether a relation between two variables represents a function. Use function notation and evaluate functions. Find the domains of functions. Use functions to model and solve real-life problems. Evaluate difference quotients. Wh ou should learn it Man natural phenomena can be modeled b functions, such as the force of water against the face of a dam, eplored in Eercise 85 on page 8. Definition of a Function A function f from a set A to a set B is a relation that assigns to each element in the set A eactl one element in the set B. The set A is the domain (or set of inputs) of the function f, and the set B contains the range (or set of outputs). To help understand this definition, look at the function that relates the time of da to the temperature in Figure.. Time of da (P.M.) Temperature (in degrees C) 9 5 Set A is the domain. Set B contains the range. Inputs:,,,, 5, Outputs: 9, 0,,, Kunio Owaki/Corbis Figure. This function can be represented b the ordered pairs, 9,,,, 5,, 5, 5,,, 0. In each ordered pair, the first coordinate (-value) is the input and the second coordinate (-value) is the output. Characteristics of a Function from Set A to Set B. Each element of A must be matched with an element of B.. Some elements of B ma not be matched with an element of A.. Two or more elements of A ma be matched with the same element of B.. An element of A (the domain) cannot be matched with two different elements of B. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

17 Section. Functions 7 Librar of Functions: Data Defined Function Man functions do not have simple mathematical formulas, but are defined b real-life data. Such functions arise when ou are using collections of data to model real-life applications. Functions can be represented in four was.. Verball b a sentence that describes how the input variables are related to the output variables Eample: The input value is the election ear from 95 to 00 and the output value is the elected president of the United States.. Numericall b a table or a list of ordered pairs that matches input values with output values Eample: In the set of ordered pairs,,, 0,, 5, 8, 50, 0, 5, the input value is the age of a male child in ears and the output value is the height of the child in inches.. Graphicall b points on a graph in a coordinate plane in which the input values are represented b the horizontal ais and the output values are represented b the vertical ais Eample: See Figure.5.. Algebraicall b an equation in two variables Eample: The formula for temperature, F 9 5C, where F is the temperature in degrees Fahrenheit and C is the temperature in degrees Celsius, is an equation that represents a function. You will see that it is often convenient to approimate data using a mathematical model or formula. STUDY TIP To determine whether or not a relation is a function, ou must decide whether each input value is matched with eactl one output value. If an input value is matched with two or more output values, the relation is not a function. Eample Testing for Functions Decide whether the relation represents as a function of. a. b. Input, 5 Output, Figure.5 a. This table does not describe as a function of. The input value is matched with two different -values. b. The graph in Figure.5 does describe as a function of. Each input value is matched with eactl one output value. Now tr Eercise 5. Prerequisite Skills When plotting points in a coordinate plane, the -coordinate is the directed distance from the -ais to the point, and the -coordinate is the directed distance from the -ais to the point. To review point plotting, see Appendi B.. STUDY TIP Be sure ou see that the range of a function is not the same as the use of range relating to the viewing window of a graphing utilit. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

18 8 Chapter Functions and Their Graphs In algebra, it is common to represent functions b equations or formulas involving two variables. For instance, the equation represents the variable as a function of the variable. In this equation, is the independent variable and is the dependent variable. The domain of the function is the set of all values taken on b the independent variable, and the range of the function is the set of all values taken on b the dependent variable. Eample Testing for Functions Represented Algebraicall Which of the equations represent(s) as a function of? a. b. To determine whether is a function of, tr to solve for in terms of. a. Solving for ields Write original equation.. Solve for. Each value of corresponds to eactl one value of. So, is a function of. b. Solving for ields Write original equation. Add to each side. ±. Solve for. The ± indicates that for a given value of there correspond two values of. For instance, when, or. So, is not a function of. Function Notation Now tr Eercise 9. When an equation is used to represent a function, it is convenient to name the function so that it can be referenced easil. For eample, ou know that the equation describes as a function of. Suppose ou give this function the name f. Then ou can use the following function notation. Input Output Equation f f The smbol f is read as the value of f at or simpl f of. The smbol f corresponds to the -value for a given. So, ou can write f. Keep in mind that f is the name of the function, whereas f is the output value of the function at the input value. In function notation, the input is the independent variable and the output is the dependent variable. For instance, the function f has function values denoted b f, f 0, and so on. To find these values, substitute the specified input values into the given equation. For, f 5. For 0, f Eploration Use a graphing utilit to graph. Then use the graph to write a convincing argument that each -value has at most one -value. Use a graphing utilit to graph. (Hint: You will need to use two equations.) Does the graph represent as a function of? Eplain. TECHNOLOGY TIP You can use a graphing utilit to evaluate a function. Go to this tetbook s Online Stud Center and use the Evaluating an Algebraic Epression program. The program will prompt ou for a value of, and then evaluate the epression in the equation editor for that value of. Tr using the program to evaluate several different functions of. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

19 Section. Functions 9 Although f is often used as a convenient function name and is often used as the independent variable, ou can use other letters. For instance, f 7, f t t t 7, all define the same function. In fact, the role of the independent variable is that of a placeholder. Consequentl, the function could be written as f 7. Eample Evaluating a Function and g s s s 7 Let g. Find (a) g, (b) g t, and (c) g. a. Replacing with in g ields the following. g 8 5 b. Replacing with t ields the following. g t t t t t c. Replacing with ields the following. g Substitute for. 8 Multipl. 8 5 Distributive Propert Simplif. Now tr Eercise 9. In Eample, note that g is not equal to g g. In general, g u v g u g v. Librar of Parent Functions: Piecewise-Defined Function A piecewise-defined function is a function that is defined b two or more equations over a specified domain. The absolute value function given b f can be written as a piecewise-defined function. The basic characteristics of the absolute value function are summarized below. A review of piecewise-defined functions can be found in the Stud Capsules. f, Graph of, Domain:, Range: 0, Intercept: 0, 0 Decreasing on, 0 Increasing on 0, 0 < 0 f() = (0, 0) Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

20 0 Chapter Functions and Their Graphs Eample A Piecewise Defined Function Evaluate the function when and 0. f,, < 0 0 Because is less than 0, use f to obtain f. For 0, use f to obtain f 0 0. Now tr Eercise 7. TECHNOLOGY TIP Most graphing utilities can graph piecewise-defined functions. For instructions on how to enter a piecewise-defined function into our graphing utilit, consult our user s manual. You ma find it helpful to set our graphing utilit to dot mode before graphing such functions. The Domain of a Function The domain of a function can be described eplicitl or it can be implied b the epression used to define the function. The implied domain is the set of all real numbers for which the epression is defined. For instance, the function f has an implied domain that consists of all real other than ±. These two values are ecluded from the domain because division b zero is undefined. Another common tpe of implied domain is that used to avoid even roots of negative numbers. For eample, the function f Domain ecludes -values that result in division b zero. Domain ecludes -values that result in even roots of negative numbers. is defined onl for 0. So, its implied domain is the interval 0,. In general, the domain of a function ecludes values that would cause division b zero or result in the even root of a negative number. Eploration Use a graphing utilit to graph. What is the domain of this function? Then graph. What is the domain of this function? Do the domains of these two functions overlap? If so, for what values? Librar of Parent Functions: Radical Function Radical functions arise from the use of rational eponents. The most common radical function is the square root function given b f. The basic characteristics of the square root function are summarized below. A review of radical functions can be found in the Stud Capsules. Graph of f Domain: 0, Range: 0, Intercept: 0, 0 Increasing on 0, f() = (0, 0) STUDY TIP Because the square root function is not defined for < 0, ou must be careful when analzing the domains of complicated functions involving the square root smbol. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

21 Section. Functions Eample 5 Finding the Domain of a Function Find the domain of each function. a. f :, 0,,, 0,,,,, b. g 5 c. h 5 d. Volume of a sphere: V r Prerequisite Skills In Eample 5(e), 0 is a linear inequalit. To review solving of linear inequalities, see Appendi E. e. k a. The domain of f consists of all first coordinates in the set of ordered pairs. Domain,, 0,, b. The domain of g is the set of all real numbers. c. Ecluding -values that ield zero in the denominator, the domain of h is the set of all real numbers ecept 5. d. Because this function represents the volume of a sphere, the values of the radius r must be positive. So, the domain is the set of all real numbers r such that r > 0. e. This function is defined onl for -values for which 0. B solving this inequalit, ou will find that the domain of k is all real numbers that are less than or equal to. Now tr Eercise 59. In Eample 5(d), note that the domain of a function ma be implied b the phsical contet. For instance, from the equation V r, ou would have no reason to restrict r to positive values, but the phsical contet implies that a sphere cannot have a negative or zero radius. For some functions, it ma be easier to find the domain and range of the function b eamining its graph. Eample Finding the Domain and Range of a Function Use a graphing utilit to find the domain and range of the function f 9. Graph the function as 9, as shown in Figure.. Using the trace feature of a graphing utilit, ou can determine that the -values etend from to and the -values etend from 0 to. So, the domain of the function f is all real numbers such that and the range of f is all real numbers such that 0. f() = 9 Now tr Eercise. Figure. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

22 Chapter Functions and Their Graphs Applications Eample 7 Cellular Communications Emploees The number N (in thousands) of emploees in the cellular communications industr in the United States increased in a linear pattern from 998 to 00 (see Figure.7). In 00, the number dropped, then continued to increase through 00 in a different linear pattern. These two patterns can be approimated b the function N(t.5t 5.,.8t 0., where t represents the ear, with t 8 corresponding to 998. Use this function to approimate the number of emploees for each ear from 998 to 00. (Source: Cellular Telecommunications & Internet Association) From 998 to 00, use N t.5t 5..., 57.9, 8., From 00 to 00, use N t.8t , 08.0, t t Now tr Eercise 8. Number of emploees (in thousands) Cellular Communications Emploees N Year (8 998) Figure.7 t Eample 8 The Path of a Baseball A baseball is hit at a point feet above the ground at a velocit of 00 feet per second and an angle of 5. The path of the baseball is given b the function f 0.00 where and f are measured in feet. Will the baseball clear a 0-foot fence located 00 feet from home plate? Algebraic The height of the baseball is a function of the horizontal distance from home plate. When 00, ou can find the height of the baseball as follows. f 0.00 Write original function. f Substitute 00 for. 5 Simplif. When 00, the height of the baseball is 5 feet, so the baseball will clear a 0-foot fence. Now tr Eercise 85. Graphical Use a graphing utilit to graph the function Use the value feature or the zoom and trace features of the graphing utilit to estimate that 5 when 00, as shown in Figure.8. So, the ball will clear a 0-foot fence Figure.8 Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

23 Section. Functions Difference Quotients One of the basic definitions in calculus emplos the ratio f h f, h h 0. This ratio is called a difference quotient, as illustrated in Eample 9. Eample 9 Evaluating a Difference Quotient For f 7, find f h f. h f h f h Summar of Function Terminolog Function: A function is a relationship between two variables such that to each value of the independent variable there corresponds eactl one value of the dependent variable. Function Notation: f h h 7 7 h h h h 7 7 h h h h h h h h Now tr Eercise 89. h, h 0 f is the name of the function. is the dependent variable, or output value. is the independent variable, or input value. f is the value of the function at. Domain: The domain of a function is the set of all values (inputs) of the independent variable for which the function is defined. If is in the domain of f, f is said to be defined at. If is not in the domain of f, f is said to be undefined at. Range: The range of a function is the set of all values (outputs) assumed b the dependent variable (that is, the set of all function values). Implied Domain: If f is defined b an algebraic epression and the domain is not specified, the implied domain consists of all real numbers for which the epression is defined. STUDY TIP Notice in Eample 9 that h cannot be zero in the original epression. Therefore, ou must restrict the domain of the simplified epression b adding h 0 so that the simplified epression is equivalent to the original epression. The smbol in calculus. indicates an eample or eercise that highlights algebraic techniques specificall used Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

24 Chapter Functions and Their Graphs. Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check Fill in the blanks.. A relation that assigns to each element from a set of inputs, or, eactl one element in a set of outputs, or, is called a.. For an equation that represents as a function of, the variable is the set of all in the domain, and the variable is the set of all in the range.. The function f, 0 is an eample of a function., > 0. If the domain of the function f is not given, then the set of values of the independent variable for which the epression is defined is called the. f h f 5. In calculus, one of the basic definitions is that of a, given b, h 0. h In Eercises, does the relation describe a function? Eplain our reasoning.. Domain Range. Domain Range. Input, 0 0 Output, 0. Domain Range. Domain Cubs (Year) National Pirates League Dodgers In Eercises 5 8, decide whether the relation represents as a function of. Eplain our reasoning American League Orioles Yankees Twins Input, 0 5 Range (Number of North Atlantic tropical storms and hurricanes) Input, Output, 9 5 Input, Output, In Eercises 9 and 0, which sets of ordered pairs represent functions from A to B? Eplain. 9. A 0,,, and B,, 0,, (a) 0,,,,, 0,, (b) 0,,,,,,, 0,, (c) 0, 0,, 0,, 0,, 0 (d) 0,,, 0,, 0. A a, b, c and B 0,,, (a) a,, c,, c,, b, (b) a,, b,, c, (c), a, 0, a,, c,, b (d) c, 0, b, 0, a, Output, Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

25 Section. Functions 5 Circulation of Newspapers In Eercises and, use the graph, which shows the circulation (in millions) of dail newspapers in the United States. (Source: Editor & Publisher Compan). Is the circulation of morning newspapers a function of the ear? Is the circulation of evening newspapers a function of the ear? Eplain.. Let f represent the circulation of evening newspapers in ear. Find f 00. In Eercises, determine whether the equation represents as a function of In Eercises 5 and, fill in the blanks using the specified function and the given values of the independent variable. Simplif the result. 5. Circulation (in millions) f (a) f (c) f t. g (a) (b) (c) (d) (b) (d) g g g t g c Morning Evening Year f 0 f c In Eercises 7, evaluate the function at each specified value of the independent variable and simplif. 7. f t t (a) f (b) f (c) f t 8. g 7 (a) g 0 (b) g 7 (c) g s 9. h t t t (a) h (b) h.5 (c) h 0. V r r (a) V (b) V (c) V r. f (a) f (b) f 0.5 (c) f. f 8 (a) f 8 (b) f (c) f 8. q 9 (a) q 0 (b) q (c) q. q t t t (a) q (b) q 0 (c) q 5. (a) f (b) f (c) f t. f (a) f (b) f (c) f t, < 0 7. f, 0 (a) f (b) f 0 (c) f 8. f f 5,, 0 > 0 (a) f (b) f 0 (c) f 9. f,, > (a) f (b) f (c) f 0. f, 0, > 0 (a) f (b) f 0 (c) f, < 0. f, 0 <, (a) f (b) f (c) f Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

26 Chapter Functions and Their Graphs. 5, f 5,, (a) f (b) f (c) In Eercises, complete the table.. h t t < 0 0 < t 5 h t f 57. f 58. f 59. g 0. h 0. g. f 0 In Eercises, use a graphing utilit to graph the function. Find the domain and range of the function.. f. f 5.. g g In Eercises 7 50, find all real values of f f 5 8. f 5 9. f 50. f 5 7 such that In Eercises 5 and 5, find the value(s) of for which f g f s s s s 0 f s f, 0, > 0 h 9, <, f, g f, g 7 5 In Eercises 5, find the domain of the function. 5. f 5 5. g 55. h t 5. s t f 5 h In Eercises 7 70, assume that the domain of f is the set A {,, 0,, }. Determine the set of ordered pairs representing the function f. 7. f f f f 7. Geometr Write the area A of a circle as a function of its circumference C. 7. Geometr Write the area A of an equilateral triangle as a function of the length s of its sides. 7. Eploration The cost per unit to produce a radio model is $0. The manufacturer charges $90 per unit for orders of 00 or less. To encourage large orders, the manufacturer reduces the charge b $0.5 per radio for each unit ordered in ecess of 00 (for eample, there would be a charge of $87 per radio for an order size of 0). (a) The table shows the profit P (in dollars) for various numbers of units ordered,. Use the table to estimate the maimum profit. Units, Profit, P (b) Plot the points, P from the table in part (a). Does the relation defined b the ordered pairs represent P as a function of? (c) If P is a function of, write the function and determine its domain. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

27 Section. Functions 7 7. Eploration An open bo of maimum volume is to be made from a square piece of material, centimeters on a side, b cutting equal squares from the corners and turning up the sides (see figure). (a) The table shows the volume V (in cubic centimeters) of the bo for various heights (in centimeters). Use the table to estimate the maimum volume. Height, Volume, V (b) Plot the points, V from the table in part (a). Does the relation defined b the ordered pairs represent V as a function of? (c) If V is a function of, write the function and determine its domain. (d) Use a graphing utilit to plot the point from the table in part (a) with the function from part (c). How closel does the function represent the data? Eplain. 7. Geometr A rectangle is bounded b the -ais and the semicircle (see figure). Write the area A of the rectangle as a function of, and determine the domain of the function. 8 = 77. Postal Regulations A rectangular package to be sent b the U.S. Postal Service can have a maimum combined length and girth (perimeter of a cross section) of 08 inches (see figure). (, ) 75. Geometr A right triangle is formed in the first quadrant b the - and -aes and a line through the point, see figure. Write the area A of the triangle as a function of, and determine the domain of the function. (0, ) (, ) (, 0) (a) Write the volume V of the package as a function of. What is the domain of the function? (b) Use a graphing utilit to graph the function. Be sure to use an appropriate viewing window. (c) What dimensions will maimize the volume of the package? Eplain. 78. Cost, Revenue, and Profit A compan produces a to for which the variable cost is $.0 per unit and the fied costs are $98,000. The to sells for $7.98. Let be the number of units produced and sold. (a) The total cost for a business is the sum of the variable cost and the fied costs. Write the total cost C as a function of the number of units produced. (b) Write the revenue R as a function of the number of units sold. (c) Write the profit P as a function of the number of units sold. (Note: P R C. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

28 8 Chapter Functions and Their Graphs Revenue In Eercises 79 8, use the table, which shows the monthl revenue (in thousands of dollars) of a landscaping business for each month of 00, with representing Januar. Month, A mathematical model that represents the data is.97. f What is the domain of each part of the piecewise-defined function? Eplain our reasoning. 80. Use the mathematical model to find f 5. Interpret our result in the contet of the problem. 8. Use the mathematical model to find f. Interpret our result in the contet of the problem. 8. How do the values obtained from the model in Eercises 80 and 8 compare with the actual data values? 8. Motor Vehicles The numbers n (in billions) of miles traveled b vans, pickup trucks, and sport utilit vehicles in the United States from 990 to 00 can be approimated b the model n t.t 75.8t 577,.9t 7, Revenue, t < t where t represents the ear, with t 0 corresponding to 990. Use the table feature of a graphing utilit to approimate the number of miles traveled b vans, pickup trucks, and sport utilit vehicles for each ear from 990 to 00. (Source: U.S. Federal Highwa Administration) Miles traveled (in billions) n Figure for Year (0 990) 8. Transportation For groups of 80 or more people, a charter bus compan determines the rate per person according to the formula Rate n 80, n 80 where the rate is given in dollars and n is the number of people. (a) Write the revenue R of the bus compan as a function of n. (b) Use the function from part (a) to complete the table. What can ou conclude? n R(n) (c) Use a graphing utilit to graph R and determine the number of people that will produce a maimum revenue. Compare the result with our conclusion from part (b). 85. Phsics The force F (in tons) of water against the face of a dam is estimated b the function F where is the depth of the water (in feet). (a) Complete the table. What can ou conclude from it? F() (b) Use a graphing utilit to graph the function. Describe our viewing window. (c) Use the table to approimate the depth at which the force against the dam is,000,000 tons. How could ou find a better estimate? (d) Verif our answer in part (c) graphicall. t Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

29 Section. Functions 9 8. Data Analsis The graph shows the retail sales (in billions of dollars) of prescription drugs in the United States from 995 through 00. Let f represent the retail sales in ear. (Source: National Association of Chain Drug Stores) Year (a) Find f 000. f 00 f 995 (b) Find and interpret the result in the contet of the problem. (c) An approimate model for the function is P t 0.098t.5t 8.85t 9.8, 5 t where P is the retail sales (in billions of dollars) and t represents the ear, with t 5 corresponding to 995. Complete the table and compare the results with the data in the graph. (d) Use a graphing utilit to graph the model and the data in the same viewing window. Comment on the validit of the model. In Eercises 87 9, find the difference quotient and simplif our answer f, 90. Retail sales (in billions of dollars) f, f() t P(t) g, f, f c f, c 0 c g h g, h 0 h f h f, h 0 h f h f, h 0 h Snthesis True or False? In Eercises 9 and 9, determine whether the statement is true or false. Justif our answer. 9. The domain of the function f is,, and the range of f is 0,. 9. The set of ordered pairs 8,,, 0,, 0,,, 0,,, represents a function. Librar of Parent Functions In Eercises 95 98, write a piecewise-defined function for the graph shown (, ) 5 (0, ) Writing In our own words, eplain the meanings of domain and range. 00. Think About It Describe an advantage of function notation. Skills Review In Eercises 0 0, perform the operation and simplif f t t, f, (, 0) 0 (, ) (, ) (, 0) (5, ) (, ) f t f, t t f f 7, 7 7 (, 0) (0, ) (, ) 8 (0, 0) (, ) 0 (, ) (, ) 5 The smbol indicates an eample or eercise that highlights algebraic techniques specificall used in calculus. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

30 0 Chapter Functions and Their Graphs. Graphs of Functions The Graph of a Function In Section., functions were represented graphicall b points on a graph in a coordinate plane in which the input values are represented b the horizontal ais and the output values are represented b the vertical ais. The graph of a function f is the collection of ordered pairs, f such that is in the domain of f. As ou stud this section, remember the geometric interpretations of and f. the directed distance from the -ais f the directed distance from the -ais Eample shows how to use the graph of a function to find the domain and range of the function. Eample Finding the Domain and Range of a Function Use the graph of the function f shown in Figure.9 to find (a) the domain of f, (b) the function values f and f, and (c) the range of f. What ou should learn Find the domains and ranges of functions and use the Vertical Line Test for functions. Determine intervals on which functions are increasing, decreasing, or constant. Determine relative maimum and relative minimum values of functions. Identif and graph step functions and other piecewise-defined functions. Identif even and odd functions. Wh ou should learn it Graphs of functions provide a visual relationship between two variables. For eample, in Eercise 88 on page 0, ou will use the graph of a step function to model the cost of sending a package. Range (, ) = f ( ) (, 0) 5 Stephen Chernin/Gett Images Figure.9 Domain a. The closed dot at, 5 indicates that is in the domain of f, whereas the open dot at, 0 indicates that is not in the domain. So, the domain of f is all in the interval,. b. Because, 5 is a point on the graph of f, it follows that f 5. Similarl, because, is a point on the graph of f, it follows that f. c. Because the graph does not etend below f 5or above f, the range of f is the interval 5,. Now tr Eercise. STUDY TIP The use of dots (open or closed) at the etreme left and right points of a graph indicates that the graph does not etend beond these points. If no such dots are shown, assume that the graph etends beond these points. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

31 Section. Graphs of Functions Eample Finding the Domain and Range of a Function Find the domain and range of f. Algebraic Because the epression under a radical cannot be negative, the domain of f is the set of all real numbers such that 0. Solve this linear inequalit for as follows. (For help with solving linear inequalities, see Appendi E.) 0 Write original inequalit. Add to each side. So, the domain is the set of all real numbers greater than or equal to. Because the value of a radical epression is never negative, the range of f is the set of all nonnegative real numbers. Graphical Use a graphing utilit to graph the equation, as shown in Figure.0. Use the trace feature to determine that the -coordinates of points on the graph etend from to the right. When is greater than or equal to, the epression under the radical is nonnegative. So, ou can conclude that the domain is the set of all real numbers greater than or equal to. From the graph, ou can see that the -coordinates of points on the graph etend from 0 upwards. So ou can estimate the range to be the set of all nonnegative real numbers. 5 = 8 Now tr Eercise 7. Figure.0 B the definition of a function, at most one -value corresponds to a given -value. It follows, then, that a vertical line can intersect the graph of a function at most once. This leads to the Vertical Line Test for functions. Vertical Line Test for Functions A set of points in a coordinate plane is the graph of as a function of if and onl if no vertical line intersects the graph at more than one point. Eample Vertical Line Test for Functions Use the Vertical Line Test to decide whether the graphs in Figure. represent as a function of. 8 a. This is not a graph of as a function of because ou can find a vertical line that intersects the graph twice. b. This is a graph of as a function of because ever vertical line intersects the graph at most once. Now tr Eercise 7. (a) 7 (b) Figure. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

32 Chapter Functions and Their Graphs Increasing and Decreasing Functions The more ou know about the graph of a function, the more ou know about the function itself. Consider the graph shown in Figure.. Moving from left to right, this graph falls from to 0, is constant from 0 to, and rises from to. Increasing, Decreasing, and Constant Functions A function f is increasing on an interval if, for an and in the interval, < implies f < f. A function f is decreasing on an interval if, for an and in the interval, < implies f > f. A function f is constant on an interval if, for an and in the interval, f f. TECHNOLOGY TIP Most graphing utilities are designed to graph functions of more easil than other tpes of equations. For instance, the graph shown in Figure.(a) represents the equation 0. To use a graphing utilit to duplicate this graph ou must first solve the equation for to obtain ±, and then graph the two equations and in the same viewing window. Eample Increasing and Decreasing Functions In Figure., determine the open intervals on which each function is increasing, decreasing, or constant. a. Although it might appear that there is an interval in which this function is constant, ou can see that if then < <,, which implies that f < f. So, the function is increasing over the entire real line. b. This function is increasing on the interval,, decreasing on the interval,, and increasing on the interval,. c. This function is increasing on the interval, 0, constant on the interval 0,, and decreasing on the interval,. Decreasing Constant Increasing Figure. f() = (, ) f() = f() = +, < 0, 0 + > (0, ) (, ) (, ) (a) Figure. (b) (c) Now tr Eercise. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

33 Relative Minimum and Maimum Values The points at which a function changes its increasing, decreasing, or constant behavior are helpful in determining the relative maimum or relative minimum values of the function. Section. Graphs of Functions Definitions of Relative Minimum and Relative Maimum A function value f a is called a relative minimum of f if there eists an interval, that contains a such that < < implies f a f. A function value f a is called a relative maimum of f if there eists an interval, that contains a such that < < implies f a f. Relative maima Relative minima Figure. shows several different eamples of relative minima and relative maima. In Section., ou will stud a technique for finding the eact points at which a second-degree polnomial function has a relative minimum or relative maimum. For the time being, however, ou can use a graphing utilit to find reasonable approimations of these points. Figure. Eample 5 Approimating a Relative Minimum Use a graphing utilit to approimate the relative minimum of the function given b f. The graph of f is shown in Figure.5. B using the zoom and trace features of a graphing utilit, ou can estimate that the function has a relative minimum at the point 0.7,.. See Figure.. Later, in Section., ou will be able to determine that the eact point at which the relative minimum occurs is, 0. f() = 5.8 TECHNOLOGY TIP When ou use a graphing utilit to estimate the - and -values of a relative minimum or relative maimum, the zoom feature will often produce graphs that are nearl flat, as shown in Figure.. To overcome this problem, ou can manuall change the vertical setting of the viewing window. The graph will stretch verticall if the values of Ymin and Yma are closer together Figure.5 Figure. Now tr Eercise. TECHNOLOGY TIP Some graphing utilities have built-in programs that will find minimum or maimum values. These features are demonstrated in Eample. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

34 Chapter Functions and Their Graphs Eample Approimating Relative Minima and Maima Use a graphing utilit to approimate the relative minimum and relative maimum of the function given b f. The graph of f is shown in Figure.7. B using the zoom and trace features or the minimum and maimum features of the graphing utilit, ou can estimate that the function has a relative minimum at the point 0.58, 0.8 See Figure.8. and a relative maimum at the point 0.58, 0.8. See Figure.9. If ou take a course in calculus, ou will learn a technique for finding the eact points at which this function has a relative minimum and a relative maimum. Eample 7 Now tr Eercise. Temperature During a -hour period, the temperature (in degrees Fahrenheit) of a certain cit can be approimated b the model , where represents the time of da, with 0 corresponding to A.M. Approimate the maimum and minimum temperatures during this -hour period. 0 To solve this problem, graph the function as shown in Figure.0. Using the zoom and trace features or the maimum feature of a graphing utilit, ou can determine that the maimum temperature during the -hour period was approimatel F. This temperature occurred at about : P.M.., as shown in Figure.. Using the zoom and trace features or the minimum feature, ou can determine that the minimum temperature during the -hour period was approimatel F, which occurred at about :8 A.M. 9.8, as shown in Figure.. f() = + Figure.7 f() = + Figure.8 f() = + Figure.9 TECHNOLOGY SUPPORT For instructions on how to use the minimum and maimum features, see Appendi A; for specific kestrokes, go to this tetbook s Online Stud Center. = Figure.0 Figure. Figure. Now tr Eercise Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

35 Graphing Step Functions and Piecewise-Defined Functions Section. Graphs of Functions 5 Librar of Parent Functions: Greatest Integer Function The greatest integer function, denoted b and defined as the greatest integer less than or equal to, has an infinite number of breaks or steps one at each integer value in its domain. The basic characteristics of the greatest integer function are summarized below. A review of the greatest integer function can be found in the Stud Capsules. Graph of f Domain:, Range: the set of integers -intercepts: in the interval 0, -intercept: 0, 0 Constant between each pair of consecutive integers Jumps verticall one unit at each integer value Could ou describe the greatest integer function using a piecewise-defined function? How does the graph of the greatest integer function differ from the graph of a line with a slope of zero? f() = [[ ]] TECHNOLOGY TIP Most graphing utilities displa graphs in connected mode, which means that the graph has no breaks. When ou are sketching graphs that do have breaks, it is better to use dot mode. Graph the greatest integer function [often called Int ] in connected and dot modes, and compare the two results. Because of the vertical jumps described above, the greatest integer function is an eample of a step function whose graph resembles a set of stairsteps. Some values of the greatest integer function are as follows. greatest integer 0 greatest integer greatest integer.5 In Section., ou learned that a piecewise-defined function is a function that is defined b two or more equations over a specified domain. To sketch the graph of a piecewise-defined function, ou need to sketch the graph of each equation on the appropriate portion of the domain. Eample 8 Sketch the graph of Graphing a Piecewise-Defined Function f,, b hand. > This piecewise-defined function is composed of two linear functions. At and to the left of, the graph is the line given b. To the right of, the graph is the line given b (see Figure.). Notice that the point, 5 is a solid dot and the point, is an open dot. This is because f 5. Now tr Eercise. Figure. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

36 Chapter Functions and Their Graphs Even and Odd Functions A graph has smmetr with respect to the -ais if whenever, is on the graph, so is the point,. A graph has smmetr with respect to the origin if whenever, is on the graph, so is the point,. A graph has smmetr with respect to the -ais if whenever, is on the graph, so is the point,. A function whose graph is smmetric with respect to the -ais is an even function. A function whose graph is smmetric with respect to the origin is an odd function. A graph that is smmetric with respect to the -ais is not the graph of a function ecept for the graph of 0. These three tpes of smmetr are illustrated in Figure.. (, ) (, ) (, ) (, ) (, ) (, ) Smmetric to -ais Smmetric to origin Smmetric to -ais Even function Odd function Not a function Figure. Test for Even and Odd Functions A function f is even if, for each in the domain of f, f f. A function f is odd if, for each in the domain of f, f f. Eample 9 Testing for Evenness and Oddness Is the function given b f even, odd, or neither? Algebraic This function is even because f f. Graphical Use a graphing utilit to enter in the equation editor, as shown in Figure.5. Then graph the function using a standard viewing window, as shown in Figure.. You can see that the graph appears to be smmetric about the -ais. So, the function is even. 0 = 0 0 Now tr Eercise 59. Figure.5 Figure. 0 Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

37 Section. Graphs of Functions 7 Eample 0 Even and Odd Functions Determine whether each function is even, odd, or neither. a. g b. h c. f Algebraic a. This function is odd because g g. b. This function is even because h h. c. Substituting for produces f. Because f and f, ou can conclude that f f and f f. So, the function is neither even nor odd. Graphical a. In Figure.7, the graph is smmetric with respect to the origin. So, this function is odd. (, ) Figure.7 b. In Figure.8, the graph is smmetric with respect to the -ais. So, this function is even. (, ) (, ) h() = + Figure.8 (, ) g() = c. In Figure.9, the graph is neither smmetric with respect to the origin nor with respect to the -ais. So, this function is neither even nor odd. f() = Now tr Eercise. Figure.9 To help visualize smmetr with respect to the origin, place a pin at the origin of a graph and rotate the graph 80. If the result after rotation coincides with the original graph, the graph is smmetric with respect to the origin. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

38 8 Chapter Functions and Their Graphs. Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check Fill in the blanks.. The graph of a function f is a collection of, such that is in the domain of f.. The is used to determine whether the graph of an equation is a function of in terms of.. A function f is on an interval if, for an and in the interval, < implies f > f.. A function value f a is a relative of f if there eists an interval, containing a such that < < implies f a f. 5. The function f is called the function, and is an eample of a step function.. A function f is if, for each in the domain of f, f f. In Eercises, use the graph of the function to find the domain and range of f. Then find f 0... = f().. In Eercises 5 0, use a graphing utilit to graph the function and estimate its domain and range. Then find the domain and range algebraicall. 5. f. f 7. f 8. h t t = f() f f 5 In Eercises, use the given function to answer the questions. (a) Determine the domain of the function. (b) Find the value(s) of such that f 0. = f() 5 = f() (c) The values of from part (b) are referred to as what graphicall? (d) Find f 0, if possible. (e) The value from part (d) is referred to as what graphicall? (f) What is the value of f at? What are the coordinates of the point? (g) What is the value of f at? What are the coordinates of the point? (h) The coordinates of the point on the graph of f at which. can be labeled, f or,.. f. f.. f() = In Eercises 5 8, use the Vertical Line Test to determine whether is a function of. Describe how ou can use a graphing utilit to produce the given graph. 5.. f() = +, 0, > 0 8 Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

39 Section. Graphs of Equations In Eercises 9, determine the open intervals over which the function is increasing, decreasing, or constant. In Eercises 0, (a) use a graphing utilit to graph the function and (b) determine the open intervals on which the function is increasing, decreasing, or constant.. f. f 5. f. f 7. f In Eercises, use a graphing utilit to approimate an relative minimum or relative maimum values of the function.. f. f h. g f 0. f. f. f f f f In Eercises 7, (a) approimate the relative minimum or relative maimum values of the function b sketching its graph using the point-plotting method, (b) use a graphing utilit to approimate an relative minimum or relative maimum values, and (c) compare our answers from parts (a) and (b). 7. f 5 8. f 9. f 0. f. f. f 8 In Eercises 50, sketch the graph of the piecewisedefined function b hand , 0 7. f, 0 <, > 5, 8. g, < < 5, Librar of Parent Functions In Eercises 5 5, sketch the graph of the function b hand. Then use a graphing utilit to verif the graph. 5. f 5. f 5. f 5. f 55. f 5. f In Eercises 57 and 58, use a graphing utilit to graph the function. State the domain and range of the function. Describe the pattern of the graph. 57. f,, f,, f,, f,, f,, h,, s < 0 0 > < 0 0 > < 0 0 > 58. g Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

40 0 Chapter Functions and Their Graphs In Eercises 59, algebraicall determine whether the function is even, odd, or neither. Verif our answer using a graphing utilit. 59. f t t t 0. f. g 5. h 5. f. f 5 5. g s s. f s s Think About It In Eercises 7 7, find the coordinates of a second point on the graph of a function f if the given point is on the graph and the function is (a) even and (b) odd. 7., 8. 5, 7 9., , 7., 7. a, c In Eercises 7 8, use a graphing utilit to graph the function and determine whether it is even, odd, or neither. Verif our answer algebraicall. 7. f 5 7. f f 7. f h 78. f f 80. g t t 8. f 8. f 5 In Eercises 8 8, graph the function and determine the interval(s) (if an) on the real ais for which f ~ 0. Use a graphing utilit to verif our results. 8. f 8. f 85. f 9 8. f 87. Communications The cost of using a telephone calling card is $.05 for the first minute and $0.8 for each additional minute or portion of a minute. (a) A customer needs a model for the cost C of using the calling card for a call lasting t minutes. Which of the following is the appropriate model? C t t C t t (b) Use a graphing utilit to graph the appropriate model. Use the value feature or the zoom and trace features to estimate the cost of a call lasting 8 minutes and 5 seconds. 88. Deliver Charges The cost of sending an overnight package from New York to Atlanta is $9.80 for a package weighing up to but not including pound and $.50 for each additional pound or portion of a pound. Use the greatest integer function to create a model for the cost C of overnight deliver of a package weighing pounds, where > 0. Sketch the graph of the function. In Eercises 89 and 90, write the height h of the rectangle as a function of = + (, ) h h (, ) (, ) = 9. Population During a ear period from 990 to 00, the population P (in thousands) of West Virginia fluctuated according to the model P 0.008t 0.t 0.0t 7.9t 79, 0 t where t represents the ear, with t 0 corresponding to 990. (Source: U.S. Census Bureau) (a) Use a graphing utilit to graph the model over the appropriate domain. (b) Use the graph from part (a) to determine during which ears the population was increasing. During which ears was the population decreasing? (c) Approimate the maimum population between 990 and Fluid Flow The intake pipe of a 00-gallon tank has a flow rate of 0 gallons per minute, and two drain pipes have a flow rate of 5 gallons per minute each. The graph shows the volume V of fluid in the tank as a function of time t. Determine in which pipes the fluid is flowing in specific subintervals of the one-hour interval of time shown on the graph. (There are man correct answers.) Volume (in gallons) V (0, 75) (0, 75) (5, 50) (5, 50) (0, 00) (0, 5) (0, 5) (0, 0) (50, 50) Time (in minutes) t Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

41 Section. Graphs of Equations Snthesis True or False? In Eercises 9 and 9, determine whether the statement is true or false. Justif our answer. 9. A function with a square root cannot have a domain that is the set of all real numbers. 9. It is possible for an odd function to have the interval 0, as its domain. Think About It In Eercises 95 00, match the graph of the function with the best choice that describes the situation. (a) The air temperature at a beach on a sunn da (b) The height of a football kicked in a field goal attempt (c) The number of children in a famil over time (d) The population of California as a function of time (e) The depth of the tide at a beach over a -hour period (f) The number of cupcakes on a tra at a part Proof Prove that a function of the following form is odd. a n n a n n... a a 0. Proof Prove that a function of the following form is even. 0. If f is an even function, determine if g is even, odd, or neither. Eplain. (a) g f (b) g f (c) g f (d) g f 0. Think About It Does the graph in Eercise represent as a function of? Eplain. 05. Think About It Does the graph in Eercise 7 represent as a function of? Eplain. 0. Writing Write a short paragraph describing three different functions that represent the behaviors of quantities between 995 and 00. Describe one quantit that decreased during this time, one that increased, and one that was constant. Present our results graphicall. Skills Review In Eercises 07 0, identif the terms. Then identif the coefficients of the variable terms of the epression In Eercises, find (a) the distance between the two points and (b) the midpoint of the line segment joining the points.., 7,,... 5, 0,, 5,,,,,, In Eercises 5 8, evaluate the function at each specified value of the independent variable and simplif. 5. f 5 (a) f (b) f (c) f. f (a) f (b) f (c) f 7. f (a) f (b) f (c) f 8. f (a) f (b) f 0 (c) f In Eercises 9 and 0, find the difference quotient and simplif our answer. 9. f 9, f h f, h h 0 f h f 0. f 5,, h 0 h a n n a n n... a a 0 Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

42 Chapter Functions and Their Graphs. Shifting, Reflecting, and Stretching Graphs Summar of Graphs of Parent Functions One of the goals of this tet is to enable ou to build our intuition for the basic shapes of the graphs of different tpes of functions. For instance, from our stud of lines in Section., ou can determine the basic shape of the graph of the linear function f m b. Specificall, ou know that the graph of this function is a line whose slope is m and whose -intercept is 0, b. The si graphs shown in Figure.0 represent the most commonl used functions in algebra. Familiarit with the basic characteristics of these simple graphs will help ou analze the shapes of more complicated graphs. f() = c f() = What ou should learn Recognize graphs of parent functions. Use vertical and horizontal shifts and reflections to graph functions. Use nonrigid transformations to graph functions. Wh ou should learn it Recognizing the graphs of parent functions and knowing how to shift, reflect, and stretch graphs of functions can help ou sketch a wide variet of simple functions b hand.this skill is useful in sketching graphs of functions that model real-life data. For eample, in Eercise 57 on page 9, ou are asked to sketch a function that models the amount of fuel used b vans, pickups, and sport utilit vehicles from 990 through 00. (a) Constant Function (b) Identit Function f() = f() = Tim Bole/Gett Images 5 (c) Absolute Value Function (d) Square Root Function f() = f() = (e) Quadratic Function Figure.0 ( f ) Cubic Function Throughout this section, ou will discover how man complicated graphs are derived b shifting, stretching, shrinking, or reflecting the parent graphs shown above. Shifts, stretches, shrinks, and reflections are called transformations. Man graphs of functions can be created from combinations of these transformations. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

43 Section. Shifting, Reflecting, and Stretching Graphs Vertical and Horizontal Shifts Man functions have graphs that are simple transformations of the graphs of parent functions summarized in Figure.0. For eample, ou can obtain the graph of h b shifting the graph of f two units upward, as shown in Figure.. In function notation, h and f are related as follows. h Upward shift of two units Similarl, ou can obtain the graph of b shifting the graph of f two units to the right, as shown in Figure.. In this case, the functions g and f have the following relationship. h() = + f g g f 5 (, ) (, ) f() = Right shift of two units f() = g() = ( ), Figure. Vertical shift upward: Figure. Horizontal shift to the two units right: two units ( ( 5 (, ( Eploration Use a graphing utilit to displa (in the same viewing window) the graphs of c, where c, 0,, and. Use the results to describe the effect that c has on the graph. Use a graphing utilit to displa (in the same viewing window) the graphs of c, where c, 0,, and. Use the results to describe the effect that c has on the graph. The following list summarizes vertical and horizontal shifts. Vertical and Horizontal Shifts Let c be a positive real number. Vertical and horizontal shifts in the graph of f are represented as follows.. Vertical shift c units upward:. Vertical shift c units downward:. Horizontal shift c units to the right: h f c h f c h f c. Horizontal shift c units to the left: h f c In items and, be sure ou see that h f c corresponds to a right shift and h f c corresponds to a left shift for c > 0. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

44 Chapter Functions and Their Graphs Eample Shifts in the Graph of a Function Compare the graph of each function with the graph of f. a. g b. h c. k a. Graph f and g [see Figure.(a)]. You can obtain the graph of g b shifting the graph of f one unit downward. b. Graph f and h [see Figure.(b)]. You can obtain the graph of h b shifting the graph of f one unit to the right. c. Graph f and k [see Figure.(c)]. You can obtain the graph of k b shifting the graph of f two units to the left and then one unit upward. (, ) g() = (, 0) (, ) f() = (, ) 5 (, ) f() = (, ) f() = h() = ( ) k() = ( + ) + (a) Vertical shift: one unit downward Figure. (b) Horizontal shift: one unit right (c) Two units left and one unit upward Now tr Eercise. Eample Finding Equations from Graphs The graph of f is shown in Figure.. Each of the graphs in Figure.5 is a transformation of the graph of f. Find an equation for each function. f() = = g() = h() (a) Figure. Figure.5 (b) a. The graph of g is a vertical shift of four units upward of the graph of f. So, the equation for g is g. b. The graph of h is a horizontal shift of two units to the left, and a vertical shift of one unit downward, of the graph of f. So, the equation for h is h. Now tr Eercise 7. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

45 Section. Shifting, Reflecting, and Stretching Graphs 5 Reflecting Graphs Another common tpe of transformation is called a reflection. For instance, if ou consider the -ais to be a mirror, the graph of h is the mirror image (or reflection) of the graph of f (see Figure.). f() = h() = Eploration Compare the graph of each function with the graph of f b using a graphing utilit to graph the function and f in the same viewing window. Describe the transformation. a. g b. h Figure. Reflections in the Coordinate Aes Reflections in the coordinate aes of the graph of f are represented as follows.. Reflection in the -ais: h f. Reflection in the -ais: h f Eample Finding Equations from Graphs The graph of f is shown in Figure.7. Each of the graphs in Figure.8 is a transformation of the graph of f. Find an equation for each function. f() = 5 = g() = h() Figure.7 (a) Figure.8 (b) a. The graph of g is a reflection in the -ais followed b an upward shift of two units of the graph of f. So, the equation for g is g. b. The graph of h is a horizontal shift of three units to the right followed b a reflection in the -ais of the graph of f. So, the equation for h is h. Now tr Eercise 9. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

46 Chapter Functions and Their Graphs Eample Reflections and Shifts Compare the graph of each function with the graph of f. a. g b. h c. k Algebraic a. Relative to the graph of f, the graph of g is a reflection in the -ais because g f. b. The graph of h is a reflection of the graph of f in the -ais because h f. c. From the equation Graphical a. Use a graphing utilit to graph f and g in the same viewing window. From the graph in Figure.9, ou can see that the graph of g is a reflection of the graph of f in the -ais. b. Use a graphing utilit to graph f and h in the same viewing window. From the graph in Figure.50, ou can see that the graph of h is a reflection of the graph of f in the -ais. c. Use a graphing utilit to graph f and k in the same viewing window. From the graph in Figure.5, ou can see that the graph of k is a left shift of two units of the graph of f, followed b a reflection in the -ais. f() = h() = f() = k f ou can conclude that the graph of k is a left shift of two units, followed b a reflection in the -ais, of the graph of f. 8 g() = Figure.9 Figure.50 f() = Now tr Eercise. Figure.5 k() = + When graphing functions involving square roots, remember that the domain must be restricted to eclude negative numbers inside the radical. For instance, here are the domains of the functions in Eample. Domain of g : 0 Domain of h : 0 Domain of k : Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

47 Nonrigid Transformations Horizontal shifts, vertical shifts, and reflections are called rigid transformations because the basic shape of the graph is unchanged. These transformations change onl the position of the graph in the coordinate plane. Nonrigid transformations are those that cause a distortion a change in the shape of the original graph. For instance, a nonrigid transformation of the graph of f is represented b cf, where the transformation is a vertical stretch if c > and a vertical shrink if 0 < c <. Another nonrigid transformation of the graph of f is represented b h f c, where the transformation is a horizontal shrink if c > and a horizontal stretch if 0 < c <. Section. Shifting, Reflecting, and Stretching Graphs 7 Eample 5 Nonrigid Transformations Compare the graph of each function with the graph of f. a. b. h g a. Relative to the graph of the graph of h f is a vertical stretch (each -value is multiplied b ) of the graph of f. (See Figure.5.) b. Similarl, the graph of g f f, is a vertical shrink each -value is multiplied b of the graph of f. (See Figure.5.) Now tr Eercise. f() = Figure.5 f() = g() = Figure h() = (, ) (, ) (, ) (, ( Eample Nonrigid Transformations Compare the graph of h f with the graph of f. Relative to the graph of f, the graph of h f 8 is a horizontal stretch (each -value is multiplied b ) of the graph of f. (See Figure.5.) Now tr Eercise 9. h() = 8 Figure.5 (, ) (, ) f() = Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

48 8 Chapter Functions and Their Graphs. Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check In Eercises 5, fill in the blanks.. The graph of a is U-shaped.. The graph of an is V-shaped.. Horizontal shifts, vertical shifts, and reflections are called.. A reflection in the -ais of f is represented b h, while a reflection in the -ais of f is represented b h. 5. A nonrigid transformation of f represented b cf is a vertical stretch if and a vertical shrink if.. Match the rigid transformation of f with the correct representation, where c > 0. (a) h f c (i) horizontal shift c units to the left (b) h f c (ii) vertical shift c units upward (c) h f c (iii) horizontal shift c units to the right (d) h f c (iv) vertical shift c units downward In Eercises, sketch the graphs of the three functions b hand on the same rectangular coordinate sstem. Verif our result with a graphing utilit.. f. f g h. f. f g g h h 5. f. f g g h h 7. f 8. f g g h h f g h f. f g h g h f g h g h. Use the graph of f to sketch each graph. To print an enlarged cop of the graph, go to the website (a) f (b) f (c) f (d) f (e) f (f) (g) f f. Use the graph of f to sketch each graph. To print an enlarged cop of the graph, go to the website (a) f (b) f (c) f (d) f (e) f (f) f (g) f (, ) f (, ) (, 0) (0, ) (, ) f (0, ) (, 0) (, ) Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

49 Section. Shifting, Reflecting, and Stretching Graphs 9 In Eercises 5 0, identif the parent function and describe the transformation shown in the graph. Write an equation for the graphed function In Eercises, compare the graph of the function with the graph of f In Eercises 7, compare the graph of the function with the graph of f In Eercises 8, compare the graph of the function with the graph of f.. g. g 5. h. h 7. p 8. p In Eercises 9, use a graphing utilit to graph the three functions in the same viewing window. Describe the graphs of g and h relative to the graph of f. 9. f 0. f g f g f h f h f f. f. g f g f h f h f g In Eercises 5, g is related to one of the si parent functions on page. (a) Identif the parent function f. (b) Describe the sequence of transformations from f to g. (c) Sketch the graph of g b hand. (d) Use function notation to write g in terms of the parent function f.. g g. g 7. g 8. g 9. g 50. g g 9 5. g 5. g 55. g 5. g 57. Fuel Use The amounts of fuel F (in billions of gallons) used b vans, pickups, and SUVs (sport utilit vehicles) from 990 through 00 are shown in the table. A model for the data can be approimated b the function F t.0. t, where t 0 represents 990. (Source: U.S. Federal Highwa Administration) Year Annual fuel use, F (in billions of gallons) (a) Describe the transformation of the parent function f t t. (b) Use a graphing utilit to graph the model and the data in the same viewing window. (c) Rewrite the function so that t 0 represents 00. Eplain how ou got our answer. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

50 50 Chapter Functions and Their Graphs 58. Finance The amounts M (in billions of dollars) of home mortgage debt outstanding in the United States from 990 through 00 can be approimated b the function M t.t 79 where t 0 represents 990. (Source: Board of Governors of the Federal Reserve Sstem) (a) Describe the transformation of the parent function f t t. (b) Use a graphing utilit to graph the model over the interval 0 t. (c) According to the model, when will the amount of debt eceed 0 trillion dollars? (d) Rewrite the function so that t 0 represents 000. Eplain how ou got our answer. Snthesis True or False? In Eercises 59 and 0, determine whether the statement is true or false. Justif our answer. 59. The graph of f is a reflection of the graph of f in the -ais. 0. The graph of f is a reflection of the graph of f in the -ais.. Eploration Use the fact that the graph of f has -intercepts at and to find the -intercepts of the given graph. If not possible, state the reason. (a) f (b) f (c) f (d) f (e) f. Eploration Use the fact that the graph of f has -intercepts at and to find the -intercepts of the given graph. If not possible, state the reason. (a) f (b) f (c) f (d) f (e) f. Eploration Use the fact that the graph of f is increasing on the interval, and decreasing on the interval, to find the intervals on which the graph is increasing and decreasing. If not possible, state the reason. (a) f (b) f (c) f (d) f (e) f. Eploration Use the fact that the graph of f is increasing on the intervals, and, and decreasing on the interval, to find the intervals on which the graph is increasing and decreasing. If not possible, state the reason. (a) f (b) (c) f (d) f (e) f Librar of Parent Functions In Eercises 5 8, determine which equation(s) ma be represented b the graph shown. There ma be more than one correct answer. 5.. (a) (a) (b) (b) f (c) f (c) f (d) f (d) f (e) f (e) f (f) f (f) f (a) (b) (c) (d) (e) (f) f f f f (a) f f (b) f f (c) f f (d) f f (e) f f (f) f Skills Review In Eercises 9 and 70, determine whether the lines L and L passing through the pairs of points are parallel, perpendicular, or neither. 9. L :,,, 0 L :,,, L :, 7,, L :, 5,, 7 In Eercises 7 7, find the domain of the function f 7. f f f Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

51 Section.5 Combinations of Functions 5.5 Combinations of Functions Arithmetic Combinations of Functions Just as two real numbers can be combined b the operations of addition, subtraction, multiplication, and division to form other real numbers, two functions can be combined to create new functions. If f and g, ou can form the sum, difference, product, and quotient of f and g as follows. f g f g f g f g, ± Sum Difference Product Quotient The domain of an arithmetic combination of functions f and g consists of all real numbers that are common to the domains of f and g. In the case of the quotient f g, there is the further restriction that g 0. What ou should learn Add, subtract, multipl, and divide functions. Find compositions of one function with another function. Use combinations of functions to model and solve real-life problems. Wh ou should learn it Combining functions can sometimes help ou better understand the big picture. For instance, Eercises 75 and 7 on page 0 illustrate how to use combinations of functions to analze U.S. health care ependitures. SuperStock Sum, Difference, Product, and Quotient of Functions Let f and g be two functions with overlapping domains. Then, for all common to both domains, the sum, difference, product, and quotient of f and g are defined as follows.. Sum:. Difference:. Product: f g f g f g f g fg f g f g f. Quotient: g 0 g, Eample Finding the Sum of Two Functions Given f and g, find f g. Then evaluate the sum when. f g f g When, the value of this sum is f g. Now tr Eercise 7(a). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

52 5 Chapter Functions and Their Graphs Eample Finding the Difference of Two Functions Given f and g, find f g. Then evaluate the difference when. Algebraic The difference of the functions f and g is f g f g. When, the value of this difference is f g. Note that f g can also be evaluated as follows. f g f g 5 7 Now tr Eercise 7(b). Graphical You can use a graphing utilit to graph the difference of two functions. Enter the functions as follows (see Figure.55). Graph as shown in Figure.5. Then use the value feature or the zoom and trace features to estimate that the value of the difference when is. Figure.55 Figure.5 5 = + In Eamples and, both f and g have domains that consist of all real numbers. So, the domain of both f g and f g is also the set of all real numbers. Remember that an restrictions on the domains of f or g must be considered when forming the sum, difference, product, or quotient of f and g. For instance, the domain of f is all 0, and the domain of g is 0,. This implies that the domain of f g is 0,. Eample Finding the Product of Two Functions Given f and g, find fg. Then evaluate the product when. fg f g When, the value of this product is fg. Now tr Eercise 7(c). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

53 Section.5 Combinations of Functions 5 Eample Finding the Quotient of Two Functions Find f g and g f for the functions given b f and g. Then find the domains of f g and g f. The quotient of f and g is g f f g and the quotient of g and f is g g f f The domain of f is 0, and the domain of g is,. The intersection of these domains is 0,. So, the domains for f g and g f are as follows. Domain of f g : 0, TECHNOLOGY TIP You can confirm the domain of f g in Eample with our graphing utilit b entering the three functions,, and, and graphing, as shown in Figure.57. Use the trace feature to determine that the -coordinates of points on the graph etend from 0 to but do not include. So, ou can estimate the domain of f g to be 0,. You can confirm the domain of g f in Eample b entering and graphing, as shown in Figure.58. Use the trace feature to determine that the -coordinates of points on the graph etend from 0 to but do not include 0. So, ou can estimate the domain of g f to be 0,. Compositions of Functions Another wa of combining two functions is to form the composition of one with the other. For instance, if f and g, the composition of f with g is f g f. This composition is denoted as f g and is read as f composed with g. Definition of Composition of Two Functions The composition of the function f with the function g is f g f g.,. Now tr Eercise 7(d). Domain of g f : 0, The domain of f g is the set of all in the domain of g such that g is in the domain of f. (See Figure.59.) f = ( () = g 5 Figure.57 g = ( () = f 5 Figure.58 f g g() g f Domain of g Domain of f Figure.59 ( ( f(g()) Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

54 5 Chapter Functions and Their Graphs Eample 5 Forming the Composition of f with g Find f g for f, 0, and g,. If possible, find f g and f g 0. f g f g Definition of f g f, Definition of g Definition of f The domain of f g is,. So, f g is defined, but f g 0 is not defined because 0 is not in the domain of f g. Now tr Eercise 5. The composition of f with g is generall not the same as the composition of g with f. This is illustrated in Eample. Eploration Let f and g. Are the compositions f g and g f equal? You can use our graphing utilit to answer this question b entering and graphing the following functions. What do ou observe? Which function represents f g and which represents g f? Eample Compositions of Functions Given f and g, evaluate (a) f g and (b) g f when 0,,, and. Algebraic a. f g f g f Definition of f g Definition of g Definition of f f g 0 0 f g 5 f g f g b. g f g f () g Definition of g f Definition of f Definition of g g f g f 5 g f g f Note that f g g f. Now tr Eercise 7. Numerical a. You can use the table feature of a graphing utilit to evaluate f g when 0,,, and. Enter g and f g in the equation editor (see Figure.0). Then set the table to ask mode to find the desired function values (see Figure.). Finall, displa the table, as shown in Figure.. b. You can evaluate g f when 0,,, and b using a procedure similar to that of part (a). You should obtain the table shown in Figure.. Figure.0 Figure. Figure. Figure. From the tables ou can see that f g g f. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

55 Section.5 Combinations of Functions 55 To determine the domain of a composite function f g, ou need to restrict the outputs of g so that the are in the domain of f. For instance, to find the domain of f g given that f and g, consider the outputs of g. These can be an real number. However, the domain of f is restricted to all real numbers ecept 0. So, the outputs of g must be restricted to all real numbers ecept 0. This means that g 0, or. So, the domain of f g is all real numbers ecept. Eample 7 Finding the Domain of a Composite Function Find the domain of the composition f g for the functions given b f 9 and g 9. Algebraic The composition of the functions is as follows. f g f g f From this, it might appear that the domain of the composition is the set of all real numbers. This, however, is not true. Because the domain of f is the set of all real numbers and the domain of g is,, the domain of f g is,. Now tr Eercise 9. Graphical You can use a graphing utilit to graph the composition of the functions f as 9 g 9. Enter the functions as follows. 9 Graph, as shown in Figure.. Use the trace feature to determine that the -coordinates of points on the graph etend from to. So, ou can graphicall estimate the domain of f g to be,. 0 Figure. 9 = ( 9 9 ( Eample 8 A Case in Which Given f and g, find each composition. a. f g b. g f a. f g f g f Now tr Eercise 5. f g g f b. g f g f () g STUDY TIP In Eample 8, note that the two composite functions f g and g f are equal, and both represent the identit function. That is, f g and g f. You will stud this special case in the net section. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

56 5 Chapter Functions and Their Graphs In Eamples 5,, 7, and 8, ou formed the composition of two given functions. In calculus, it is also important to be able to identif two functions that make up a given composite function. Basicall, to decompose a composite function, look for an inner and an outer function. Eample 9 Identifing a Composite Function Write the function h 5 as a composition of two functions. One wa to write h as a composition of two functions is to take the inner function to be g 5 and the outer function to be f. Then ou can write h 5 f 5 f g. Now tr Eercise 5. Eploration Write each function as a composition of two functions. h a. b. r What do ou notice about the inner and outer functions? Eample 0 Identifing a Composite Function Write the function h as a composition of two functions. One wa to write h as a composition of two functions is to take the inner function to be g and the outer function to be f Then ou can write h. f f g. Now tr Eercise 9. Eploration The function in Eample 0 can be decomposed in other was. For which of the following pairs of functions is h equal to f g? a. g and f b. g and f c. g and f Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

57 Section.5 Combinations of Functions 57 Eample Bacteria Count The number N of bacteria in a refrigerated food is given b where T is the temperature of the food (in degrees Celsius). When the food is removed from refrigeration, the temperature of the food is given b where t is the time (in hours). a. Find the composition N T t and interpret its meaning in contet. b. Find the number of bacteria in the food when t hours. c. Find the time when the bacterial count reaches 000. a. N T 0T 80T 500, T t t, 0t 0 The composite function N T t represents the number of bacteria as a function of the amount of time the food has been out of refrigeration. b. When t, the number of bacteria is N c. The bacterial count will reach N 000 when 0t You can solve this equation for t algebraicall as follows. 0t t 580 t 79 0 t t 79 t. hours So, the count will reach 000 when t. hours. When ou solve this equation, note that the negative value is rejected because it is not in the domain of the composite function. You can use a graphing utilit to confirm our solution. First graph the equation N 0t 0, as shown in Figure.5. Then use the zoom and trace features to approimate N 000 when t., as shown in Figure.. Now tr Eercise 8. T N T t 0 t 80 t t t 0t t 0t 80 0t N = 0t + 0, t Figure Figure. Eploration Use a graphing utilit to graph 0 0 and 000 in the same viewing window. (Use a viewing window in which 0 and ) Eplain how the graphs can be used to answer the question asked in Eample (c). Compare our answer with that given in part (c). When will the bacteria count reach 00? Notice that the model for this bacteria count situation is valid onl for a span of hours. Now suppose that the minimum number of bacteria in the food is reduced from 0 to 00. Will the number of bacteria still reach a level of 000 within the three-hour time span? Will the number of bacteria reach a level of 00 within hours? Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

58 58 Chapter Functions and Their Graphs.5 Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check Fill in the blanks.. Two functions f and g can be combined b the arithmetic operations of,,, and to create new functions.. The of the function f with the function g is f g f g.. The domain of f g is the set of all in the domain of g such that is in the domain of f.. To decompose a composite function, look for an and an function. In Eercises, use the graphs of f and g to graph h f g. To print an enlarged cop of the graph, go to the website f g g f.. In Eercises 5, find (a) f g, (b) f g, (c) fg, and (d) f/g. What is the domain of f/g? 5.. f, f 5, g g 7. f, g f 5, f 5, f, g g g.. 5 f, f, f g g g g f In Eercises, evaluate the indicated function for f and g algebraicall. If possible, use a graphing utilit to verif our answer.. f g. f g 5. f g 0. f g 7. fg 8. fg f g t. f g t. fg 5t. fg t 5.. In Eercises 7 0, use a graphing utilit to graph the functions f, g, and h in the same viewing window f, f, g, g, h f g h f g 9. f, g, h f g 0. f, g, h f g In Eercises, use a graphing utilit to graph f, g, and f g in the same viewing window. Which function contributes most to the magnitude of the sum when 0 } }? Which function contributes most to the magnitude of the sum when >?.. f g 5 f g t f, f, g 0 g f g 0. f, g 5. f, g g f t Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

59 Section.5 Combinations of Functions 59 In Eercises 5 8, find (a) f g, (b) g f, and, if possible, (c) f g In Eercises 9 8, determine the domains of (a) f, (b) g, and (c) f g. Use a graphing utilit to verif our results f, g f, g f 5, g 5 f g, f, g f, g() f, f 5, In Eercises, use the graphs of f and g to evaluate the functions. = f( ) g g = g( )... f, g f, g f g,. (a) f g (b). (a) f g (b). (a) f g (b). (a) f g (b) f g fg g f g f f, f, f, f, f, g g g g g In Eercises 9 5, (a) find f g, g f, and the domain of f g. (b) Use a graphing utilit to graph f g and g f. Determine whether f g g f f, f, f, g g g f, f, f, g g g In Eercises 55 0, (a) find f g and g f, (b) determine algebraicall whether f g g f, and (c) use a graphing utilit to complete a table of values for the two compositions to confirm our answers to part (b) f 5, f, f, f, g g g 5 g 0 In Eercises 5 7, find two functions f and g such that f g h. (There are man correct answers.) 5. h. h 7. h 8. h h h 5 7. h 7. h 7. Stopping Distance The research and development department of an automobile manufacturer has determined that when required to stop quickl to avoid an accident, the distance (in feet) a car travels during the driver s reaction time is given b R where is the speed of the car in miles per hour. The distance (in feet) traveled while the driver is braking is given b B 5. (a) Find the function that represents the total stopping distance T. (b) Use a graphing utilit to graph the functions R, B, and T in the same viewing window for 0 0. (c) Which function contributes most to the magnitude of the sum at higher speeds? Eplain. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

60 0 Chapter Functions and Their Graphs 7. Sales From 000 to 00, the sales (in thousands of dollars) for one of two restaurants owned b the same parent compan can be modeled b R 80 8t 0.8t, for t 0,,,,, 5,, where t 0 represents 000. During the same seven-ear period, the sales R (in thousands of dollars) for the second restaurant can be modeled b R t, for t 0,,,,, 5,. (a) Write a function R that represents the total sales for the two restaurants. (b) Use a graphing utilit to graph R, R, and R (the total sales function) in the same viewing window. Data Analsis In Eercises 75 and 7, use the table, which shows the total amounts spent (in billions of dollars) on health services and supplies in the United States and Puerto Rico for the ears 995 through 005. The variables,, and represent out-of-pocket paments, insurance premiums, and other tpes of paments, respectivel. (Source: U.S. Centers for Medicare and Medicaid Services) The models for this data are.t 8, and.0t.t 8.t 0,.8t 7, where t represents the ear, with t 5 corresponding to Use the models and the table feature of a graphing utilit to create a table showing the values of,, and for each ear from 995 to 005. Compare these values with the original data. Are the models a good fit? Eplain. 7. Use a graphing utilit to graph,,, and T in the same viewing window. What does the function T represent? Eplain. 77. Ripples A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius (in feet) of the outermost ripple is given b r t 0.t, where t is the time (in seconds) after the pebble strikes the water. The area of the circle is given b A r r. Find and interpret A r t. R Year Geometr A square concrete foundation was prepared as a base for a large clindrical gasoline tank (see figure). (a) Write the radius r of the tank as a function of the length of the sides of the square. (b) Write the area A of the circular base of the tank as a function of the radius r. (c) Find and interpret A r. 79. Cost The weekl cost C of producing units in a manufacturing process is given b C The number of units produced in t hours is t 50t. (a) Find and interpret C t. (b) Find the number of units produced in hours. (c) Use a graphing utilit to graph the cost as a function of time. Use the trace feature to estimate (to two-decimalplace accurac) the time that must elapse until the cost increases to $5, Air Traffic Control An air traffic controller spots two planes at the same altitude fling toward each other. Their flight paths form a right angle at point P. One plane is 50 miles from point P and is moving at 50 miles per hour. The other plane is 00 miles from point P and is moving at 50 miles per hour. Write the distance s between the planes as a function of time t. Distance (in m iles) P s Distance (in miles) r Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

61 Section.5 Combinations of Functions 8. Bacteria The number of bacteria in a refrigerated food product is given b N T 0T 0T 00, for T 0, where T is the temperature of the food in degrees Celsius. When the food is removed from the refrigerator, the temperature of the food is given b T t t, where t is the time in hours. (a) Find the composite function N T t or N T t and interpret its meaning in the contet of the situation. (b) Find N T and interpret its meaning. (c) Find the time when the bacteria count reaches Pollution The spread of a contaminant is increasing in a circular pattern on the surface of a lake. The radius of the contaminant can be modeled b r t 5.5 t, where r is the radius in meters and t is time in hours since contamination. (a) Find a function that gives the area A of the circular leak in terms of the time t since the spread began. (b) Find the size of the contaminated area after hours. (c) Find when the size of the contaminated area is 50 square meters. 8. Salar You are a sales representative for an automobile manufacturer. You are paid an annual salar plus a bonus of % of our sales over $500,000. Consider the two functions f 500,000 and g() 0.0. If is greater than $500,000, which of the following represents our bonus? Eplain. (a) f g (b) g f 8. Consumer Awareness The suggested retail price of a new car is p dollars. The dealership advertised a factor rebate of $00 and an 8% discount. (a) Write a function R in terms of p giving the cost of the car after receiving the rebate from the factor. (b) Write a function S in terms of p giving the cost of the car after receiving the dealership discount. (c) Form the composite functions R S p and S R p and interpret each. (d) Find R S 8,00 and S R 8,00. Which ields the lower cost for the car? Eplain. Snthesis True or False? In Eercises 85 and 8, determine whether the statement is true or false. Justif our answer. 85. If f and g, then f g g f. 8. If ou are given two functions f and g, ou can calculate f g if and onl if the range of g is a subset of the domain of f. Eploration In Eercises 87 and 88, three siblings are of three different ages. The oldest is twice the age of the middle sibling, and the middle sibling is si ears older than one-half the age of the oungest. 87. (a) Write a composite function that gives the oldest sibling s age in terms of the oungest. Eplain how ou arrived at our answer. (b) If the oldest sibling is ears old, find the ages of the other two siblings. 88. (a) Write a composite function that gives the oungest sibling s age in terms of the oldest. Eplain how ou arrived at our answer. (b) If the oungest sibling is two ears old, find the ages of the other two siblings. 89. Proof Prove that the product of two odd functions is an even function, and that the product of two even functions is an even function. 90. Conjecture Use eamples to hpothesize whether the product of an odd function and an even function is even or odd. Then prove our hpothesis. 9. Proof Given a function f, prove that g is even and h is odd, where g f f and h f f. 9. (a) Use the result of Eercise 9 to prove that an function can be written as a sum of even and odd functions. (Hint: Add the two equations in Eercise 9.) (b) Use the result of part (a) to write each function as a sum of even and odd functions. f g, Skills Review In Eercises 9 9, find three points that lie on the graph of the equation. (There are man correct answers.) In Eercises 97 00, find an equation of the line that passes through the two points. 97.,,, 8 98., 5, 8, ,.,,.,,, Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

62 Chapter Functions and Their Graphs. Inverse Functions Inverse Functions Recall from Section. that a function can be represented b a set of ordered pairs. For instance, the function f from the set A,,, to the set B 5,, 7, 8 can be written as follows. f :, 5,,,, 7,, 8 In this case, b interchanging the first and second coordinates of each of these ordered pairs, ou can form the inverse function of f, which is denoted b f. It is a function from the set B to the set A, and can be written as follows. f : 5,,,, 7,, 8, Note that the domain of f is equal to the range of f, and vice versa, as shown in Figure.7. Also note that the functions f and f have the effect of undoing each other. In other words, when ou form the composition of f with f or the composition of f with f, ou obtain the identit function. f f f f f f What ou should learn Find inverse functions informall and verif that two functions are inverse functions of each other. Use graphs of functions to decide whether functions have inverse functions. Determine if functions are one-to-one. Find inverse functions algebraicall. Wh ou should learn it Inverse functions can be helpful in further eploring how two variables relate to each other. For eample, in Eercises 0 and 0 on page 7, ou will use inverse functions to find the European shoe sizes from the corresponding U.S. shoe sizes. Domain of f f() = + Range of f f() Range of f f () = Domain of f LWA-Dann Tardif/Corbis Figure.7 Eample Finding Inverse Functions Informall Find the inverse function of f(). Then verif that both f f and f f are equal to the identit function. The function f multiplies each input b. To undo this function, ou need to divide each input b. So, the inverse function of f is given b f. You can verif that both f f and f f are equal to the identit function as follows. f f f Now tr Eercise. f f f STUDY TIP Don t be confused b the use of the eponent to denote the inverse function f. In this tet, whenever f is written, it alwas refers to the inverse function of the function f and not to the reciprocal of f, which is given b f. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

63 Section. Inverse Functions Eample Finding Inverse Functions Informall Find the inverse function of f. Then verif that both f f and f f are equal to the identit function. The function f subtracts from each input. To undo this function, ou need to add to each input. So, the inverse function of f is given b f. You can verif that both f f and f f are equal to the identit function as follows. f f f f f f Now tr Eercise. A table of values can help ou understand inverse functions. For instance, the following table shows several values of the function in Eample. Interchange the rows of this table to obtain values of the inverse function. 0 f() f 0 In the table at the left, each output is less than the input, and in the table at the right, each output is more than the input. The formal definition of an inverse function is as follows. Definition of Inverse Function Let f and g be two functions such that f g for ever in the domain of g and g f for ever in the domain of f. Under these conditions, the function g is the inverse function of the function f. The function g is denoted b f (read f-inverse ). So, f f and f f. The domain of f must be equal to the range of f, and the range of f must be equal to the domain of f. If the function g is the inverse function of the function f, it must also be true that the function f is the inverse function of the function g. For this reason, ou can sa that the functions f and g are inverse functions of each other. Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

64 Chapter Functions and Their Graphs Eample Verifing Inverse Functions Algebraicall Show that the functions are inverse functions of each other. f and f g f g f g Eample Now tr Eercise 5. g Verifing Inverse Functions Algebraicall Which of the functions is the inverse function of g 5 or h 5 B forming the composition of f with g, ou have f g f Because this composition is not equal to the identit function, it follows that g is not the inverse function of f. B forming the composition of f with h, ou have 5 f h f So, it appears that h is the inverse function of f. You can confirm this b showing that the composition of h with f is also equal to the identit function. Now tr Eercise 9. f 5? TECHNOLOGY TIP Most graphing utilities can graph in two was: or However, ou ma not be able to obtain the complete graph of b entering. If not, ou should use. or. = 5 5 = / Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

65 Section. Inverse Functions 5 The Graph of an Inverse Function The graphs of a function f and its inverse function f are related to each other in the following wa. If the point a, b lies on the graph of f, then the point b, a must lie on the graph of f, and vice versa. This means that the graph of f is a reflection of the graph of f in the line, as shown in Figure.8. ( a, b) = f ( ) = = f ( ) TECHNOLOGY TIP In Eamples and, inverse functions were verified algebraicall. A graphing utilit can also be helpful in checking whether one function is the inverse function of another function. Use the Graph Reflection Program found at this tetbook s Online Stud Center to verif Eample graphicall. ( b, a) Figure.8 Eample 5 Verifing Inverse Functions Graphicall and Numericall Verif that the functions f and g from Eample are inverse functions of each other graphicall and numericall. Graphical You can verif that f and g are inverse functions of each other graphicall b using a graphing utilit to graph f and g in the same viewing window. (Be sure to use a square setting.) From the graph in Figure.9, ou can verif that the graph of g is the reflection of the graph of f in the line. g() = + = f() = Numerical You can verif that f and g are inverse functions of each other numericall. Begin b entering the compositions f g and g f into a graphing utilit as follows. f g g f Then use the table feature of the graphing utilit to create a table, as shown in Figure.70. Note that the entries for,, and are the same. So, f g and g f. You can now conclude that f and g are inverse functions of each other. Figure.9 Now tr Eercise 5. Figure.70 Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

66 Chapter Functions and Their Graphs The Eistence of an Inverse Function Consider the function f. The first table at the right is a table of values for f. The second table was created b interchanging the rows of the first table. The second table does not represent a function because the input is matched with two different outputs: and. So, f does not have an inverse function. To have an inverse function, a function must be one-to-one, which means that no two elements in the domain of f correspond to the same element in the range of f. Definition of a One-to-One Function A function f is one-to-one if, for a and b in its domain, f a f b implies that a b. 0 f() 0 0 g() 0 Eistence of an Inverse Function A function f has an inverse function f if and onl if f is one-to-one. = From its graph, it is eas to tell whether a function of is one-to-one. Simpl check to see that ever horizontal line intersects the graph of the function at most once. This is called the Horizontal Line Test. For instance, Figure.7 shows the graph of. On the graph, ou can find a horizontal line that intersects the graph twice. Two special tpes of functions that pass the Horizontal Line Test are those that are increasing or decreasing on their entire domains.. If f is increasing on its entire domain, then f is one-to-one.. If f is decreasing on its entire domain, then f is one-to-one. Eample Testing for One-to-One Functions Is the function f one-to-one? (, ) Figure.7 one-to-one. (, ) f is not Algebraic Let a and b be nonnegative real numbers with f a f b. a b Set f a f b. a b a b So, f a f b implies that a b. You can conclude that f is one-to-one and does have an inverse function. Graphical Use a graphing utilit to graph the function. From Figure.7, ou can see that a horizontal line will intersect the graph at most once and the function is increasing. So, f is one-to-one and does have an inverse function. = Now tr Eercise 55. Figure.7 Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

67 Section. Inverse Functions 7 Finding Inverse Functions Algebraicall For simple functions, ou can find inverse functions b inspection. For more complicated functions, however, it is best to use the following guidelines. Finding an Inverse Function The function f with an implied domain of all real numbers ma not pass the Horizontal Line Test. In this case, the domain of f ma be restricted so that f does have an inverse function. For instance, if the domain of f is restricted to the nonnegative real numbers, then f does have an inverse function. Eample 7 Finding an Inverse Function Algebraicall Find the inverse function of f. Use the Horizontal Line Test to decide whether f has an inverse function.. In the equation for f, replace f b.. Interchange the roles of and, and solve for.. Replace b f in the new equation. 5. Verif that f and f are inverse functions of each other b showing that the domain of f is equal to the range of f, the range of f is equal to the domain of f, and f f and f f. The graph of f in Figure.7 passes the Horizontal Line Test. So ou know that f is one-to-one and has an inverse function. f 5 5 f Write original function. Replace f b. Interchange and. Multipl each side b. Isolate the -term. Solve for. The domains and ranges of f and f f and f f. Replace b f. f Now tr Eercise consist of all real numbers. Verif that TECHNOLOGY TIP Man graphing utilities have a built-in feature for drawing an inverse function. To see how this works, consider the function f. The inverse function of f is given b f, 0. Enter the function. Then graph it in the standard viewing window and use the draw inverse feature. You should obtain the figure below, which shows both f and its inverse function f. For instructions on how to use the draw inverse feature, see Appendi A; for specific kestrokes, go to this tetbook s Online Stud Center. 0 0 f () = 5 Figure.7 f () =, f() = f() = 5 Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

68 8 Chapter Functions and Their Graphs Eample 8 Finding an Inverse Function Algebraicall Find the inverse function of f and use a graphing utilit to graph f and f in the same viewing window. f f Write original function. Replace f b. Interchange and. Isolate. Solve for. Replace b f. The graph of f in Figure.7 passes the Horizontal Line Test. So, ou know that f is one-to-one and has an inverse function. The graph of f in Figure.7 is the reflection of the graph of f in the line. Verif that f f and f f. Now tr Eercise. f () = Figure.7 8 = f() = Eample 9 Finding an Inverse Function Algebraicall Find the inverse function of f and use a graphing utilit to graph f and f in the same viewing window. f f, 0 Write original function. Replace f b. Interchange and. Square each side. Isolate. Solve for. Replace b f. The graph of f in Figure.75 passes the Horizontal Line Test. So ou know that f is one-to-one and has an inverse function. The graph of in Figure.75 is the reflection of the graph of f in the line. Note that the range of f is the interval 0,, which implies that the domain of f is the interval 0,. Moreover, the domain of f is the interval,, which implies that the range of f is the interval,. Verif that f f and f f. Now tr Eercise 5. f f () = +, 0 5 (0, (, 0 7 Figure.75 ( = ( f() = Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

69 Section. Inverse Functions 9. Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check Fill in the blanks.. If the composite functions f g and g f, then the function g is the function of f, and is denoted b.. The domain of f is the of f, and the of f is the range of f. f. The graphs of f and are reflections of each other in the line.. To have an inverse function, a function f must be ; that is, f a f b implies a b. 5. A graphical test for the eistence of an inverse function is called the Line Test. In Eercises 8, find the inverse function of f informall. Verif that f f and f f.. f. f. f 7. f 5. f. 7. f 8. f 5 In Eercises 9, (a) show that f and g are inverse functions algebraicall and (b) use a graphing utilit to create a table of values for each function to numericall show that f and g are inverse functions f 7, f 9, f 5, g 7 g 9 g 5 f 8. f 9, 9. f, 0. In Eercises, match the graph of the function with the graph of its inverse function. [The graphs of the inverse functions are labeled (a), (b), (c), and (d).] (a) (c) f, 7 0; g 0; 9 g 9 g, 0 < (b) (d) 7 9. f, g.. f 8, f 0, g 8, g In Eercises 5 0, show that f and g are inverse functions algebraicall. Use a graphing utilit to graph f and g in the same viewing window. Describe the relationship between the graphs f, g. 7. f g, f ; g, 0 9 Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

70 70 Chapter Functions and Their Graphs In Eercises 5 8, show that f and g are inverse functions (a) graphicall and (b) numericall. 5. f, f 5, f 5, f, g g 5 5 g g 5.. h f In Eercises 7 58, determine algebraicall whether the function is one-to-one. Verif our answer graphicall. 7. f 8. g 9. f f 5 In Eercises 9, determine if the graph is that of a function. If so, determine if the function is one-to-one f f, q 5, f f f, f h.. In Eercises 59 8, find the inverse function of f algebraicall. Use a graphing utilit to graph both f and f in the same viewing window. Describe the relationship between the graphs f 0. f. f 5. f. f 5. f, 0 5. f, 0. f, 0 In Eercises 5, use a graphing utilit to graph the function and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function eists. 5. f. f 7. h 8. g 9. h 0. f. f 0. f 0.5. g 5. f f 8. f Think About It In Eercises 9 78, restrict the domain of the function f so that the function is one-to-one and has an inverse function. Then find the inverse function f. State the domains and ranges of f and f. Eplain our results. (There are man correct answers.) 9. f 70. f 7. f 7. f 7. f 7. f 75. f 5 7. f 77. f 78. f Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

71 Section. Inverse Functions 7 In Eercises 79 and 80, use the graph of the function f to complete the table and sketch the graph of f f In Eercises 8 88, use the graphs of f and g to evaluate the function. f = f() = g() 8. f 0 8. g 0 8. f g 8. g f 85. f g 0 8. g f 87. g f 88. f g Graphical Reasoning In Eercises 89 9, (a) use a graphing utilit to graph the function, (b) use the draw inverse feature of the graphing utilit to draw the inverse of the function, and (c) determine whether the graph of the inverse relation is an inverse function, eplaining our reasoning. 89. f 90. h 9. g 9. f 5 In Eercises 9 98, use the functions f 8 and g to find the indicated value or function. 9. f g 9. g f 0 f f 95. f f 9. g g 97. f g 98. g f In Eercises 99 0, use the functions f and g 5 to find the specified function. 99. g f 00. f g 0. f g 0. g f 0. Shoe Sizes The table shows men s shoe sizes in the United States and the corresponding European shoe sizes. Let f represent the function that gives the men s European shoe size in terms of, the men s U.S. size. Men s U.S. shoe size (a) Is f one-to-one? Eplain. (b) Find f. (c) Find f, if possible. (d) Find f f. (e) Find f f. 0. Shoe Sizes The table shows women s shoe sizes in the United States and the corresponding European shoe sizes. Let g epresent the function that gives the women s European shoe size in terms of, the women s U.S. size. (a) Is g one-to-one? Eplain. (b) Find g. (c) Find g ). (d) Find g g 9. (e) Find g g 5. Men s European shoe size Women s U.S. shoe size Women s European shoe size Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

72 7 Chapter Functions and Their Graphs 05. Transportation The total values of new car sales f (in billions of dollars) in the United States from 995 through 00 are shown in the table. The time (in ears) is given b t, with t 5 corresponding to 995. (Source: National Automobile Dealers Association) (a) Does f eist? (b) If f eists, what does it mean in the contet of the problem? (c) If f eists, find f 50.. (d) If the table above were etended to 005 and if the total value of new car sales for that ear were $90. billion, would f eist? Eplain. 0. Hourl Wage Your wage is $8.00 per hour plus $0.75 for each unit produced per hour. So, our hourl wage in terms of the number of units produced is (a) Find the inverse function. What does each variable in the inverse function represent? (b) Use a graphing utilit to graph the function and its inverse function. (c) Use the trace feature of a graphing utilit to find the hourl wage when 0 units are produced per hour. (d) Use the trace feature of a graphing utilit to find the number of units produced when our hourl wage is $.5. Snthesis Year, t Sales, f t True or False? In Eercises 07 and 08, determine whether the statement is true or false. Justif our answer. 07. If f is an even function, f eists. 08. If the inverse function of f eists, and the graph of f has a -intercept, the -intercept of f is an -intercept of f. 09. Proof Prove that if f and g are one-to-one functions, f g g f. 0. Proof Prove that if f is a one-to-one odd function, f is an odd function. In Eercises, decide whether the two functions shown in the graph appear to be inverse functions of each other. Eplain our reasoning..... In Eercises 5 8, determine if the situation could be represented b a one-to-one function. If so, write a statement that describes the inverse function. 5. The number of miles n a marathon runner has completed in terms of the time t in hours. The population p of South Carolina in terms of the ear t from 90 to The depth of the tide d at a beach in terms of the time t over a -hour period 8. The height h in inches of a human born in the ear 000 in terms of his or her age n in ears Skills Review In Eercises 9, write the rational epression in simplest form In Eercises 8, determine whether the equation represents as a function of Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

73 .7 Linear Models and Scatter Plots Section.7 Linear Models and Scatter Plots 7 Scatter Plots and Correlation Man real-life situations involve finding relationships between two variables, such as the ear and the outstanding household credit market debt. In a tpical situation, data is collected and written as a set of ordered pairs. The graph of such a set is called a scatter plot. (For a brief discussion of scatter plots, see Appendi B..) Eample Constructing a Scatter Plot The data in the table shows the outstanding household credit market debt D (in trillions of dollars) from 998 through 00. Construct a scatter plot of the data. (Source: Board of Governors of the Federal Reserve Sstem) What ou should learn Construct scatter plots and interpret correlation. Use scatter plots and a graphing utilit to find linear models for data. Wh ou should learn it Real-life data often follows a linear pattern. For instance, in Eercise 0 on page 8, ou will find a linear model for the winning times in the women s 00-meter freestle Olmpic swimming event. Year Household credit market debt, D (in trillions of dollars) Nick Wilson/Gett Images Begin b representing the data with a set of ordered pairs. Let t represent the ear, with t 8 corresponding to ,.0, 9,., 0, 7.0,, 7.,, 8.,, 9.,, 0. Then plot each point in a coordinate plane, as shown in Figure.7. Now tr Eercise. D Household Credit Market Debt From the scatter plot in Figure.7, it appears that the points describe a relationship that is nearl linear. The relationship is not eactl linear because the household credit market debt did not increase b precisel the same amount each ear. A mathematical equation that approimates the relationship between t and D is a mathematical model. When developing a mathematical model to describe a set of data, ou strive for two (often conflicting) goals accurac and simplicit. For the data above, a linear model of the form D at b appears to be best. It is simple and relativel accurate. Debt (in trillions of dollars) Year (8 998) Figure.7 t Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.