Math Analysis Chapter 1 Notes: Functions and Graphs

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1 Math Analysis Chapter 1 Notes: Functions and Graphs Day 6: Section 1-1 Graphs; Section 1- Basics of Functions and Their Graphs Points and Ordered Pairs The Rectangular Coordinate System (aka: The Cartesian coordinate system) Practice: Label each on the coordinate system shown. 1) x-axis, ) y-axis, 3) origin 4) Quad I, 5) Quad II, 6) Quad III, 7) Quad IV. Also plot the points: A(. 4), B(4, ), C( 3. 0), and D(0, 3). Graphing Equations by Plotting Points A relationship between two quantities can be expressed as an equations in two variables, such as: y = x 4. A solution of an equation in two variables, x and y, is an ordered pair of real numbers with the following property: When the x-coordinate is substituted for x and the y-coordinate is substituted for y in the equation, we obtain a true statement. Practice: In 1-, Determine if the given ordered pairs are solutions to the equation y = x (10, 96). (0, ) Graphing an Equation Using the Point-Plotting Method. 1. Select values for x (for this section you will use the integers from 3 to 3). Substitute each x-value into the equation and solve for y. 3. Create order pairs by grouping the x-value with it s y-value. 4. Plot the order pairs Math Analysis Notes prepared by Mr. Hopp 1

2 Practice: 1. Graph y = x 4using the Point-Plotting Method. x y = x 4 Ordered Pair (x, y) The graph of y = x 4 Plot all order pairs on a coordinate plane and connect the points with a smooth curve if and only if you are given an equation to graph (which you are). Do your homework on graph paper. If you need graph paper you can get some by printing off the net at Intercepts x-intercept of a graph is the x-coordinate when the y-coordinate equals zero. We also can describe the x-intercept graphically at the point where the graph intersects the x-axis. y-intercept of a graph is the y-coordinate when the x-coordinate equals zero. We also can describe the y-intercept graphically at the point where the graph intersect the y-axis. Practice: 1-5, Identify the x- and y-intercepts of the given graphs or equation. 1. x-int: y-int:. x-int: y-int: x 3y = 6 y = x+ 4 Math Analysis Notes prepared by Mr. Hopp

3 3. x-int: y-int: 4. x-int: y-int: 3 x + y 6x 6y+ 14= 0 y = x 3x+ 5. y = 16 x Basics of Functions and Their Graphs Relation Definition of a Relation A relation is any set of ordered pairs. The set of all x-components of the ordered pairs is called the domain of the relation and the set of all y-components is called the range of the relation. Practice: Find the domain and the range of the relation: {( 5,1.8 ),( 10,16. ),( 15,18.9 )(, 0,0.7)( 5,1.8 )} Domain: Range: Definition of a Function A function is a correspondence from a first set, called domain, to a second set, called the range, such that each element in the domain corresponds to exactly one element in the range. A relation of order pairs is a function if all of the domain (x) values are different. A function can have two different domain values with the same range value. Practice: In 1- Determine whether each relation is a function: 1. ( 1,,3,4,5,6,5,8 ) ( ) ( ) ( ). ( 1, ), ( 3, 4 ), ( 6, 5 ), ( 8, 5 ) { } { } Math Analysis Notes prepared by Mr. Hopp 3

4 Determining Whether an Equation Represents a Function. o To determine whether an equation defines y as a function of x: 1. Solve the equation in terms of y. If only one value of y can be obtained for a given x, the equation is a function. Example: x + y = 4 x x y = 4 x If two or more values of y can be obtained for a given x, the equation is not a function. Example: x + y = 5 x x y = 5 x y = ± 5 x y = ± 5 x Practice: In 1-, determine whether each equation defines y as a function of x. 1. y = x+ 4. x+ y = 4 From this last equation we can see that for each value of x, there is one and only one value of y. For example, if x = 3, then y = 4 3 = 5. This equation defines y as a function of x. The ± in the last equation shows that for certain values of x, there are two values of y. For example x =, then y = ± 5 () =± 5 4 =± 1=± 1. For this reason, the equation does not define y as a function of x. Function Notation If an equation in x and y gives only one value of y for each value of x, then the variable y is a function of the variable x. We use function notation by replacing y with f ( x ). We think of a functions domain (xcomponents) as the set of the function s input values and the range (y-components) as the set of the function s output values. The special notation f ( x ), read f of x represents the value of the function at the number x. Input Output We read this equation as f of x equal x +3 x f(x) f ( x) = x+ 3 Evaluating a Function To evaluate a function substitute the input value in for x and evaluate the expression. Practice: If f( x) = x x+ 7, evaluate each of the following: (a) f ( 5) (b) f( x+ 4) (c) f ( x) Math Analysis Notes prepared by Mr. Hopp 4

5 Graphs of Functions Practice: Graph the functions f ( x) = x and gx ( ) = x 3 in the same rectangular coordinate system. Select integers for x, starting with and ending with. How is the graph of g related to the graph of f? The Vertical Line Test If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x. If any vertical line intersects a graph in none or only at one point,, the graph does define y as a function of x. Practice: In 1-4, Use the vertical line test to identify graphs in which y is a function of x. 1.. Math Analysis Notes prepared by Mr. Hopp 5

6 3. 4. Asn 6: pg. 135#3-7 by 3 s, 4-50(even) pg. 149#-6 (even), 7-36 by 3 s, pg. 150#56-90(even) Day 7: Section 1-3 More on Functions and Their Graphs The Difference Quotient The expression: f ( x+ h) f( x) h for h 0 is called the difference quotient. To evaluate the difference quotient: 1. Find the value of f ( x+ h) by using substitution, replacing x with (x + h) and simplify the expression.. Use the expression found in step one and subtract f ( x) from it. 3. Use the expression found in step two and divide by h. This new expression is the called the difference quotient. Example: If f( x) = 3x x+ 4, find and simplify each expression. This is f(x+h) form part (a) (a) f ( x+ h) (b) f( x+ h) f( x), h 0 h (a) We find f ( x+ h) by replacing x with x + h each time that x appears in the equation. (b) Using our result from part (a), we obtain the following: f(x) = 3x x + 4 f( x+ h) f( x) 3x + xh+ h x h+ 4 ( 3x x+ 4) = h h f(x + h) = 3(x + h) 3x + xh+ h x h+ 4 3x + x 4 (x + h) + 4 = h = 3(x + xh + h ) x h + 4 ( 3x 3x ) + ( x+ x) + ( 4 4) + xh+ h h = = 3x + xh + h x h + 4 h xh+ h h = h = h( x + h ) = x + h Math Analysis Notes prepared by Mr. h Hopp 6 Replace x with x + h This is f(x) from the given equation.

7 f( x+ h) f( x) Practice: In 1-, Find and simplify the difference quotient:, h 0 for the given function. h 1 1. f( x) =. f( x) = x + 3x 1 x Piecewise Functions A function that is defined by two (or more) equations over a specified domain is called a piecewise function. To evaluate a Piecewise Function Only substitute the x-value into the expression where the inequality of x is true. When you evaluate this expression the result is the value of the function with that given x. Practice: In 1-, Evaluate each piecewise function at the given values of the independent variable. 3x+ 5 if x< 0 1. f( x) = x 9 4x+ 7 if x 0 if x 3. hx ( ) = x 3 (a) f ( ) (b) f ( 0) (c) f ( 3) 6 if x = 3 h (b) h ( 0) (c) h () 3 (a) ( 5) Increasing, Decreasing, and Constant Functions. 1. A function is increasing on an open interval, I, if for any x 1 and x in the interval where x 1 < x, then f(x 1 ) < f(x ).. A function is decreasing on an open interval, I, if for any x 1 and x in the interval where x 1 < x, then f(x 1 ) > f(x ). 3. A function is constant on an open interval, I, if for any x 1 and x in the interval where x 1 < x, then f(x 1 ) = f(x ). Increasing Decreasing Constant y y y x x x Math Analysis Notes prepared by Mr. Hopp 7

8 Practice: In 1-, State the intervals on which the given function is increasing, decreasing, or constant. 1.. Increasing: Decreasing: Constant: Increasing: Decreasing: Constant: Relative Maxima and Relative Minima Relative Maximum Definitions of Relative Maximum and Relative Minimum y 1. A function value f(a) is a relative maximum of f if there exists an open interval above a such that f( a) > f( x) for all x in the open interval. Relative Max is the point on top of a hill. A function value f(b) is a relative minimum of f if there exists an open interval about b such that f( b) < f( x) for all x in the open interval. Relative Min is the point at the bottom of a valley Relative Minimum x Practice: Use the graph of f to determine each of the following. Where applicable, use interval notation. 1. the domain of f. the range of f 3. the x-intercept(s) 4. the y-intercept(s) 5. interval(s) on which f is increasing 6. interval(s) on which f is decreasing 7. intervals(s) on which f is constant 8. the relative minimum of f 9. the relative maximum of f 10. f( 3) 11. the value(s) of x when f(x) = Math Analysis Notes prepared by Mr. Hopp 8

9 Even and Odd Functions and Symmetry The function f is an even function if: f( x) = f( x) for all x in the domain of f. The right side of the equation of an even function does not change if x is replaced with x. The function f is an odd function if: f( x) = f( x) for all x in the domain of f. The right side of the equation of an odd function changes its sign if x is replaced with x. If you replace x with x and you get something else that happens then we say the function is neither odd or even. Practice: In 1-3, Determine whether each of the following functions is even, odd, or neither f( x) = x + 6. f( x) = 7x x 3. f( x) 5 = x + 1 Even Functions and y-axis Symmetry The graph of an even function in which f( x) = f(x) is symmetric with respect to the y-axis. Odd Functions and Origin Symmetry The graph of an odd function in which f( x) = f(x) is symmetric with respect to the origin. f(x)=x 3 f(x)=x 4 Asn 7: pg. 150 #40-50 (even) pg. 163 #3-1 by 3 s, 4-64(even) Math Analysis Notes prepared by Mr. Hopp 9

10 Day 8: Section 1-4 Linear Functions and Slope; Section 1-5 More on Slope The Slope of a Line Run: x x 1 Definition of Slope The slope of the line through the distinct, x, y is: points ( x y ) and ( ) 1 1 Change in y Rise y m = = = Change in x Run x y 1 x 1 Rise: y y 1 Where x x 1 0. Practice: In 1-, Find the slope of the line passing through each pair of points: 1. ( 3, 4) and ( 4, ). (4, ) and ( 1, 5) Possibilities for a Line s Slope Positive Slope Negative Slope Zero Slope Undefined Slope m < 0 m is undefined m > 0 m = 0 Line Rises from left to right Line falls from left to right Line is horizontal Line is vertical Equations of Lines 1. Point-slope form: y y 1 = m(x x 1 ) where m = slope, (x 1, y 1 ) is a point on the line. Slope-intercept form: y = mx + b where m = slope, b = y-intercept 3. Horizontal line: y = b where b = y-intercept 4. Vertical line: x = a where a = x-intercept 5. General form: Ax + By + C = 0 where A, B and C are integers and A must be positive. Practice: In 1-6, use the given conditions to write an equation for each line in (a) point-slope form, (b) slope-intercept form and (c) General form. 1. Slope = 8, passing through (4, 1). Slope = 3, passing through 3 7 4, 8 Math Analysis Notes prepared by Mr. Hopp 10

11 3. Passing through ( 3, ) and (3, 6) 4. Passing through , and, x-intercept = 3 and y-intercept = 1 6. x-intercept = 3 and y-intercept = 5 8 Parallel and Perpendicular Lines Slope and Parallel Lines 1. If two nonvertical lines are parallel, then they have the same slope.. If two distinct nonvertical lines have the same slope, then they are parallel. 3. Two distinct vertical lines, both with undefined slopes, are parallel If two lines are parallel they have equal slopes: m 1 = m Practice: Write an equation of the line passing through (, 5) and parallel to the line whose equation is y = 3x + 1. Express the equation in point-slope from and slope-intercept form. Slope and Perpendicular Lines 1. If two nonvertical lines are perpendicular, then the product of their slopes is 1.. If the product of the slopes of two lines is 1, then the lines are perpendicular. 3. A horizontal line having zero slope is perpendicular to a vertical line having undefined slope. If two lines are perpendicular they have slopes that are negative reciprocals of each other: 1 m = m 1 Practice: 1. Find the slope of any line that is perpendicular to the line whose equation is x + 3y 1 = 0.. Write the equation of the line passing through (, 6) and perpendicular to the line whose equation is x + 3y 1 = 0. Express the equation in general form. Asn 8: pg. 179 #3-7 by 3 s pg, (even), -6 (even) Study for Quiz Sections 1-5 Math Analysis Notes prepared by Mr. Hopp 11

12 Day 9: Section 1-6 Transformations of functions Algebra s Common Graphs Constant Function f(x) = c Identity Function Absolute Value Function f(x) = x f(x) = x Domain: (, Range: the single number c, Constant on ( ) Even function (symmetric to y- axis) Domain: (, Range: (, Increasing on (, Odd function (symmetric to Origin) Domain: (, Range: [ 0, Decreasing on (,0) and Increasing on ( 0, Even function (symmetric to y- axis) Standard Quadratic Function Standard Square Root Function Standard Cubic Function f(x) = x f(x) = x 3 f(x) = x Domain: (, Range: [ 0, Decreasing on (,0) and Increasing on ( 0, Even function (symmetric to y- axis) Domain: [ 0, Range: [ 0, Increasing on ( 0, Neither even nor odd Domain: (, Range: (, Increasing on (, Odd function (symmetric to Origin) Math Analysis Notes prepared by Mr. Hopp 1

13 Standard Cube Root Function Domain: (, Range: (, Increasing on (, Odd function (symmetric to Origin Summary of Transformations (In each case, c represents a positive real number.) To Graph Draw the Graph of f and: Changes in the Equation of y=f(x) Vertical Shifts y = f(x) + c Raise the graph of f by c units c is added to f(x) y = f(x) c Horizontal Shifts y = f(x + c) y = f(x c) Reflection about the x-axis y = f(x) Reflection about the y-axis y = f( x) Vertical Stretching or Shrinking y = cf(x), c > 1 y = cf(x), 0 < c < 1 Horizontal Stretching or Shrinking y = f(cx), c > 1 y = f(cx), 0 < c < 1 Lower the graph of f by c units Shift the graph of f to the left c units Shift the graph of f to the left c units Reflect the graph of f about the x-axis Reflect the graph of f about the y-axis Multiply each y-coordinate of y = f(x) by c, vertically stretching the graph of f. Multiply each y-coordinate of y = f(x) by c, vertically shrinking the graph of f. Divide each x-coordinate of y = f(x) by c, horizontally shrinking the graph of f. Divide each x-coordinate of y = f(x) by c, horizontally stretching the graph of f. c is subtracted from f(x) x is replaced with x + c x is replaced with x c f(x) is multiplied by 1 x is replaced with x f(x) is multiplied by c, c > 1 f(x) is multiplied by c, 0 < c < 1 x is replaced with cx, c > 1 x is replaced with cx, 0 < c < 1 Math Analysis Notes prepared by Mr. Hopp 13

14 Practice: Graph the standard function f(x) and then graph the given function. Describe the transformations need to change the common function f(x) to get g(x). 1. g(x) = (x + 3) + 4. g(x) = x + 4 Description of transformations Description of transformations Asn 9: pg. 06 #8-11 by 8 s Day 10: Section 1-7 Combinations of Functions and Composite Functions; Section 1-8 Inverse Functions Finding the Domain of a Function The numbers excluded from a functions domain are real numbers that cause division by zero and real numbers that result in a square root of a negative number. Practice: In 1-3, Use interval notation to express the domain of each function: 5x 1. f( x) = x + 3x 10. f( x) = x 7 3. hx ( ) = 9x 7 Math Analysis Notes prepared by Mr. Hopp 14

15 The Algebra of Functions: Sum, Difference, Product and Quotient of Functions Let f and g be two functions. The sum f + g, the difference f g, the product fg, and the quotient f are functions whose g domains are the set of all real numbers in common to the domains of f and g defined as follows: 1. Sum: ( f + g)( x) = f( x) + g( x). Difference: ( f g)( x) = f( x) g( x) 3. Product: ( fg)( x) = f( x) g( x) 4. Quotient: f f( x) ( x) =, provided g(x) 0 g gx ( ) Practice: Let f( x) = x 5 and gx ( ) = x 1. Find each of the following functions and determine the domain: f g 1. (f + g)(x). (f g)(x) 3. (fg)(x) 4. ( x) Composite Functions The Composition of Functions The composition of the function f with g is denoted by f o g and is defined by the equation: ( f o g)( x) = f ( g( x) ). The domain of the composite function f o g is the set of all x such that 1. x is the domain of g and. g(x) is in the domain of f. Practice: Given f(x) = 5x + 6 and g(x) = x x 1, find each of the following composite functions: 1. ( f o g)( x). ( g o f )( x) 3. ( f o g)( 3) Math Analysis Notes prepared by Mr. Hopp 15

16 Inverse Functions Definition of the Inverse of a Function Let f and g be two functions such that f ( g( x) ) = x for every x in the domain of g and g f x = x for every x in the domain of f. ( ( )) The function g is the inverse of the function f and is denoted by ( ( )) 1 f f x x =. The domain of f is equal to the range of Finding the Inverse of a Function The equation for the inverse of a function f can be found as follows: 1. Replace f(x) with y in the equation for f(x).. Interchange x and y. 3. Solve for y. If this equation does not define y as a function of x, the function f does not have an inverse function and his procedure ends. If this equation does define y as a function of x, the function f has an inverse function. 4. If f has an inverse function, replace y in step 3 with f 1 ( x) f 1 f x = x 1 and ( ( )) f f x = x. Practice: Find the inverse of each function. 1. f( x) = x+ 7. f( x) 3 = 4x f 1 (read f inverse ). Thus, ( ) 1 f, and vice versa. ( ). We can verify our result by showing that ( ) f( x) 3 = 1 x ( ) f f x = x and 4. Verify that the inverse function found in problem 1 is correct. Math Analysis Notes prepared by Mr. Hopp 16

17 The Horizontal Line Test and One-to-One Functions The Horizontal Line Test For Inverse Functins A function f has an inverse that is a function f 1, if there is no horizontal line that intersects the graph of the function f at more than one point. If the function passes the Horizontal Line Test the function is said to be one-to-one. Practice: In 1-4 Which of the following graph represent functions that have inverse functions? Graph of f and f 1 To graph an inverse function given the graph of ordered pairs of the function 1. Find an ordered pair on the function.. To graph the inverse just take the x-coordinate of f(x) is the y-coordinate of f 1 (x) and the y-coordinate of f(x) is the x- coordinate of f 1 (x). 3. Continue finding order pairs on f(x) and interchange the x- and y-coordinates to plot points on f 1 (x). 4. Connect points with a smooth curve. Practice: Use the graph of f to draw the graph of its inverse function. Asn 10: pg. 06 # by 10 s (skip 80) pg. 19 #3-78 by 3 s pg. 31 #3-7 by 3 s, 9-38(all), 40, 46, 48 Math Analysis Notes prepared by Mr. Hopp 17

18 Day 11: Section 1-9 Distance and Midpoint Formulas; Circles; Section 1-10 Modeling with Functions The Distance Formula The distance, d, between the points ( x, y ) and (, ) 1 1 x y in a rectangular coordinate system is: ( ) ( ) 1 1 d = x x + y y To compute the distance between two points, find the square of the difference between the x-coordinates plus the square of the difference between the y-coordinates. The principal square root of this sum is the distance. Practice: Find the distance between the two points given. 1. ( 4, 9) and (1, 3). ( 3, 6 ) and ( 3, 5 6) The Midpoint Formula Consider a line segment whose endpoints are ( x, y ) and (, ) 1 1 x y. The coordinates of the segment s midpoint are: x1 + x y1 + y, To find the midpoint, take the average of the two x-coordinates and the average of the two y-coordinates. Practice: Find the midpoint of the line segment with endpoints at: , and, Circles Our goal is to translate a circle s geometric definition into an equation. Geometric Definition of a Circle A circle is the set of all points in a plane that are equidistant from a fixed point, called the center. The fixed distance from the circle s center to any point on the circle is called the radius. The Standard Form of the Equation of a Circle The standard form of the equation of a circle with center (h, k) and radius r is: ( ) ( ) x h + y k = r Math Analysis Notes prepared by Mr. Hopp 18

19 Practice: Write the standard form of the equation of the circle with the given information. 1. Center ( 3, 1), r = 6. Endpoints of it s diameter: (, 3) and (, 1) Converting the General Form of a Circle s Equation to Standard Form and Graphing the Circle The General Form of the Equation of a Circle x y Dx Ey F = 0 where D, E, and F are real numbers. To Convert General Form to Standard Form of a Circle 1. Group like components together and move constant term to other side.. Complete the Square for both x and y components. Remember to add the perfect square to both sides of the equal sign. 3. Factor each group on the left side of the equal sign to a square of a binomial. Your equation of a circle should now be in standard form Practice: In 1-, Write in standard form and graph: 1. x + y + 6x + y + 6 = 0. x + y + 3x y 1 = 0 Math Analysis Notes prepared by Mr. Hopp 19

20 Modeling with Functions; Word Problems Practice: 1. A car rental agency charges $00 per week plus $0.15 per mile to rent a car. (a) Express the weekly cost to rent the car, f, as a function of the number of miles driven during the week, x. (b) How many miles did you drive during the week if the weekly cost to ret the car was $30?. The bus fare in a cit is $1.5. People who use the bus have the option of purchasing a monthly coupon book for $1.00. With the coupon book, the fare is reduced to $0.50. (a) Express the total monthly cost to use the bus without a coupon book, f, as a function of the number of times in a month the bus is used, x. (b) Express the total monthly cost to use the bus with a coupon book, g, as a function of the number of times in a month the bus is used, x. (c) Determine the number of times in a moth the bus must be used so that the total monthly cost without the coupon book is the same as the total monthly cost with the coupon book. What will the monthly cost for each option? Asn 11: pg 39 #3-30 by 3 s, 3-60(evens) pg 51 #-10(evens) Math Analysis Notes prepared by Mr. Hopp 0