B Apply

59. Think About a Plan A craftsman makes and sells violins. The function I(x) = 5995 x 5995 , x  represents the income in dollars from selling x violins. The function P ( y ) = y − 100,000 p open y close equals y minus , 100,000  represents his profit in dollars if he makes an income of y dollars. What is the profit from selling 30 violins?
    • How can you write a composite function to represent the craftsman's profit?
    • How can you use the composite function to find the profit earned when he sells 30 violins?
60. Suppose your teacher offers to give the whole class a bonus if everyone passes the next math test. The teacher says she will give everyone a 10-point bonus and increase everyone's grade by 9% of their score.
    1. You earned a 75 on the test. Would you rather have the 10-point bonus first and then the 9% increase, or the 9% increase first and then the 10-point bonus?
    2. Reasoning Is this the best plan for all students? Explain.

61. Sales A salesperson earns a 3% bonus on weekly sales over $5000. Consider the following functions.

    g(x) = 0.03x

    h ( x ) = x − 5000 h open x close equals x minus , 5000

    1. Explain what each function above represents.
    2. Which composition, ( h ∘ g ) ( x ) open h composition g close open x close  or ( g ∘ h ) ( x ) , open g composition h close open x close comma  represents the weekly bonus? Explain.
62. If ( f ∘ g ) ( x ) = x 2 − 6 x + 8 open f composition g close open x close equals , x squared , minus 6 x plus 8  and g ( x ) = x − 3 , g open x close equals x minus 3 comma  what is f(x)?

Let g(x) = 3x + 2 and f ( x ) = x − 2 3 . f , open x close , equals . fraction x minus 2 , over 3 end fraction . .  Find each value.

63. f(g(1))
64. g ( f ( − 4 ) ) g open f open negative 4 close close
65. f(g(0))
66. g(f(2))
67. g(g(0))
68. ( g ∘ g ) ( 1 ) open g composition g close open 1 close
69. ( f ∘ g ) ( − 2 ) open f composition g close open negative 2 close
70. ( f ∘ f ) ( 0 ) open f composition f close open 0 close
71. Geometry You toss a pebble into a pool of water and watch the circular ripples radiate outward. You find that the function r(x) = 12.5x describes the radius r, in inches, of a circle x seconds after it was formed. The function A ( x ) = π x 2 eh open x close equals pi , x squared  describes the area A of a circle with radius x.
    1. Find ( A ∘ r ) ( x ) open eh composition r close open x close  when x = 2 . x equals 2 .  Interpret your answer.
    2. Find the area of a circle 4 seconds after it was formed.

For each pair of functions, find f(g(x)) and g(f(x)).

72. f ( x ) = 3 x , g ( x ) = x 2 f open x close equals 3 x comma g open x close equals , x squared
73. f ( x ) = x + 3 , g ( x ) = x − 5 f open x close equals x plus 3 comma g open x close equals x minus 5
74. f ( x ) = 3 x 2 + 2 , g ( x ) = 2 x f open x close equals , 3 x squared , plus 2 comma g open x close equals 2 x
75. f ( x ) = x − 3 2 , g ( x ) = 2 x − 3 f , open x close , equals . fraction x minus 3 , over 2 end fraction . comma g , open x close , equals 2 x minus 3
76. f ( x ) = − x − 7 , g ( x ) = 4 x f open x close equals negative x minus 7 comma g open x close equals 4 x
77. f ( x ) = x + 5 2 , g ( x ) = x 2 f , open x close , equals . fraction x plus 5 , over 2 end fraction . comma g , open x close , equals , x squared
78. Open-Ended Write a function rule that approximates each value.
    1. The amount you save is a percent of what you earn. (You choose the percent.)
    2. The amount you earn depends on how many hours you work. (You choose the hourly wage.)
    3. Write and simplify a composite function that expresses your savings as a function of the number of hours you work. Interpret your results.

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