2-5 Graphing Special Functions. Graph each function. Identify the domain and range. SOLUTION:

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1 Graph each function Identify the domain and range Write the piecewise-defined function shown in each graph 1 3 The left portion of the graph is the line g(x) = x + 4 There is an open circle at ( 2, 2), so the domain for this part of the function is The function is defined for all real values of x, so the domain is all real numbers The y-coordinates of points on the graph are real numbers less than or equal to 4, so the range is The center portion of the graph is the constant function g(x) = 3 There are closed dots at ( 2, 3) and (3, 3), so the domain for this part is The right portion of the graph is the line g(x) = 2x + 12 There is an open circle at (3, 6), so the domain for this part is Write the piecewise function 2 The function is defined for all real values of x, so the domain is all real numbers The y-coordinates of points on the graph are real numbers between 8 and 2 and less than or equal to 8, so the range is esolutions Manual - Powered by Cognero Page 1

2 4 The left portion of the graph is the constant function g(x) = 6 There is a closed dot at ( 5, 6), so the domain for this part of the function is 5 REASONING Springfield High School s theater can hold 250 students The drama club is performing a play in the theater Draw a graph of a step function that shows the relationship between the number of tickets sold x and the minimum number of performances y that the drama club must do When x is greater than 0 and less than or equal to 250, the drama club needs to do only one performance When x is greater than 250 and less than or equal to 500, they must do at least two performances Continue the pattern with a table The center portion of the graph is the line g(x) = x + 4 There are open circles at ( 5, 9) and ( 2, 6), so the domain for this part is The right portion of the graph is the line g(x) = There is a closed dot at ( 2, 0), so the domain for this part is Write the piecewise function esolutions Manual - Powered by Cognero Page 2

3 Graph each function Identify the domain and range 6 8 The function g(x) is a reflection of twice of a greatest integer function So, g(x) takes all even integer values or zero 9 R = {all even integers} 7 R = {all integers} 10 esolutions Manual - Powered by Cognero Page 3

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5 15 Write the piecewise-defined function shown in each graph 17 The left portion of the graph is the line g(x) = x 4 There is an open circle at ( 3, 1), so the domain for this part of the function is The center portion of the graph is the line g(x) = x + 1 There are closed dots at ( 3, 2) and (1, 2), so the domain for this part is The right portion of the graph is the constant function g(x) = 6 There is an open circle at (4, 6), so the domain for this part is Write the piecewise function 16 The left portion of the graph is the constant function g(x) = 8 There is a closed dot at ( 6, 8), so the domain for this part of the function is The center portion of the graph is the line g(x) = 025x + 2 There are closed dots at ( 4, 1) and (4, 3), so the domain for this part is The right portion of the graph is the constant function g(x) = 4 There is an open circle at (6, 4), so the constant function is defined for Write the piecewise function esolutions Manual - Powered by Cognero Page 5

6 18 The left portion of the graph is the constant function g(x) = 9 There is an open circle at ( 5, 9), so the domain for this part of the function is The center portion of the graph is the line g(x) = x + 4 There are closed dots at (0, 4) and (3, 7), so the domain for this part is The right portion of the graph is the line g(x) = x 3 There is an open circle at (7, 4), so the domain for this part is Write the piecewise function 19 The left portion of the graph is the constant function g(x) = 8 There is a closed dot at ( 1, 8), so the domain for this part is The center portion of the graph is the line g(x) = 2x There are closed dots at (4, 8) and (6, 12), so the domain for this part is The right portion of the graph is the line g(x) = 2x 15 There is a circle at (7, 1), so the domain for this part is Write the piecewise function Graph each function Identify the domain and range 20 R = {all integers} esolutions Manual - Powered by Cognero Page 6

7 21 23 R = {all integers} 22 The function is defined for all real values of x, so the domain is all real numbers The function g(x) is twice of a greatest integer function So, g(x) takes only even integer values Therefore, the range is R = {all even integers} Graph each function Identify the domain and range 24 R = {all integers} esolutions Manual - Powered by Cognero Page 7

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9 30 GIVING Patrick is donating money and volunteering his time to an organization that restores homes for the needy His employer will match his monetary donations up to $100 a Identify the type of function that models the total money received by the organization when Patrick donates x dollars b Write and graph a function for the situation a The function is composed of two distinct linear functions Therefore, it is a piecewise function b 31 SENSE-MAKING A car s speedometer reads 60 miles an hour a Write an absolute value function for the difference between the car s actual speed a and the reading on the speedometer b What is an appropriate domain for the function? Explain your reasoning c Use the domain to graph the function a The absolute value function is b Since the speed of the car cannot be negative, the appropriate domain for the function is {a a 0} c esolutions Manual - Powered by Cognero Page 9

10 32 RECREATION The charge for renting a bicycle from a rental shop for different amounts of time is shown in the table a Identify the type of function that models this situation b Write and graph a function for the situation Use each graph to write the absolute value function 33 The graph changes its direction at (0, 0) The slope of the line in the interval is 05 The slope of the line in the interval is 05 a The rent is constant in each interval Therefore, the situation is best modeled by a step function Therefore, the absolute value function is b 34 The graph changes its direction at ( 5, 4) The slope of the line in the interval is 1 The slope of the line in the interval is 1 Therefore, the absolute value function is esolutions Manual - Powered by Cognero Page 10

11 35 Graph each function Identify the domain and range ; R = {non-negative integers} 39 MULTIPLE REPRESENTATIONS Consider the following sets of absolute value functions for for for for a Graphical Graph each set of functions on a graphing calculator b Verbal Compare and contrast the graphs of the esolutions Manual - Powered by Cognero Page 11

12 functions in each set c Tabular Choose any two functions Find the slope between consecutive points on the graphs d Verbal Describe how the slopes of the two sections of an absolute value graph are related a d For f (x), the slopes are opposite when the a- values are opposites For g(x), the slopes are the same when the b-values are opposites For h(x), the slopes are opposite when the c-values are opposites For k(x), the slopes are the same when the d-values are the opposites The slope is constant for each section of the graph b For f (x), negative values of a cause the graph to be reflected in the x-axis The greater the absolute value of a, the more the graph is vertically stretched For g(x), the greater the absolute value of b, the more the graph is horizontally stretched For h(x), as c increases, the graph shifts right For k(x), as d increases, the graph shifts up 40 OPEN-ENDED Write an absolute value relation in which the domain is all nonnegative numbers and the range is all real numbers y = x c esolutions Manual - Powered by Cognero Page 12

13 41 CHALLENGES Graph 45 Which type of function is a linear function? A absolute value B constant C greatest integer D step The graph of a linear function is a straight line Of the functions listed, only the graph of a constant function is a straight line So, B is the correct answer 42 CRITIQUE ARGUMENTS Find a counterexample to the following statement and explain your reasoning 46 What is the equation of the function shown in the graph? In order to find the greatest integer function of x when x is not an integer, round x to the nearest integer 86; The greatest integer function asks for the greatest integer less than or equal to the given value; thus 8 is the greatest integer If we were to round this value to the nearest integer, we would round up to 9 43 OPEN-ENDED Write an absolute value function in which f (5) = 3 44 WRITING IN MATH Explain how piecewise functions can be used to accurately represent realworld problems Piecewise functions can be used to represent the cost of items when purchased in quantities, such as a dozen eggs The graph shows the greatest integer function either translated down 2 units or translated right 2 units So, either or Thus, A and B are correct 47 MULTI-STEP Consider the function described below a Which of the following statements are true? Choose all that apply A The range of the first part of the function and the second part of the function overlap B There are two points on this function with the esolutions Manual - Powered by Cognero Page 13

14 same domain C The number 5 is included in the domain of this function D The number 4 is not included in the range of this function E The range ends at 100 F The graph of this piecewise function would include parts of two lines G The graph of this piecewise function would include parts of three lines b Which of the following options correctly describes the range of this function? 48 Consider the following function a Which of the following statements are true? Choose all that apply A The function bends once B The function bends three times C The function has no y-intercept D The value of the function at is undefined E The value of the function at is undefined F The graph of this piecewise function would include parts of two lines G The graph of this piecewise function would include parts of three lines b What is the range of this function? c If it has one, what is the vertex of this function? c What is the y-intercept of this function? d What is true about the x-intercept of this function? A It has multiple values B It cannot be determined C It can be determined and is 2 D It can be determined and is -05 a Graphing this function, we see that C, D, and G are correct b Graphing this function, we see that A is correct c 10 The graph crosses the y-axis at (0, 10) or you can use the function 5x + 10, if d D Using the function 2x + 1 = 0, we see that x = - 05 Because this falls in the domain for this part of the piecewise function, (-05, 0) is the x-intercept a Graphing this function, we see A, D, and G are correct b Graphing this function, we see the range is {f(x) f(x) 0} c Graphing this function, we find the vertex is (2,0) esolutions Manual - Powered by Cognero Page 14