B Apply

Find the inverse of each function. Is the inverse a function?

42. f ( x ) = x 3 f open x close equals , x cubed
43. f ( x ) = x 4 f open x close equals , x to the fourth
44. f ( x ) = 2 x 2 5 + 1 f , open x close , equals . fraction 2 , x squared , over 5 end fraction . plus 1
45. f ( x ) = 1.5 x 2 − 4 f open x close equals 1.5 , x squared , minus 4
46. f ( x ) = 3 x 2 4 f , open x close , equals . fraction 3 , x squared , over 4 end fraction
47. f ( x ) = 2 x − 1 + 3 f , open x close , equals . square root of 2 x minus 1 end root . plus 3
48. Think About a Plan The velocity of the water that flows from an opening at the base of a tank depends on the height of water above the opening. The function v ( x ) = 2 g x v , open x close , equals , square root of 2 g x end root  models the velocity v in feet per second where g, the acceleration due to gravity, is about 32  ft/s 2 32 , ftslashs squared and x is the height in feet of the water. What is the depth of water when the flow is 40 ft/s, and when the flow is 20 ft/s?
    • How can you use inverse functions to help you find the answer?
    • What restrictions are on the domain of v(x)? of v − 1 ( x ) ? v super negative 1 end super , open x close question mark
49. Let f ( x ) = 3 x 2 − 4 f open x close equals , 3 x squared , minus 4  and g ( x ) = x − 2 . g open x close equals x minus 2 .  Calculate ( f ∘ g − 1 ) ( x ) open f composition , g super negative 1 end super , close open x close  for x = − 3 . x equals negative 3 .
50. Writing Explain how you can find the range of the inverse of f ( x ) = x − 1 f , open x close , equals , square root of x minus 1 end root  without finding the inverse itself.

For each function, find the inverse and the domain and range of the function and its inverse. Determine whether the inverse is a function.

51. f ( x ) = − x f , open x close , equals negative square root of x
52. f ( x ) = x + 3 f , open x close , equals square root of x plus 3
53. f ( x ) = − x + 3 f , open x close , equals . square root of negative x plus 3 end root
54. f ( x ) = x + 2 f , open x close , equals , square root of x plus 2 end root
55. f ( x ) = x 2 2 f , open x close , equals , fraction x squared , over 2 end fraction
56. f ( x ) = 1 x 2 f , open x close , equals , fraction 1 , over x squared end fraction
57. f ( x ) = ( x − 4 ) 2 f open x close equals . open x minus 4 close squared
58. f ( x ) = ( 7 − x ) 2 f open x close equals . open 7 minus x close squared
59. f ( x ) = 1 ( x + 1 ) 2 f , open x close , equals . fraction 1 , over open , x plus 1 , close squared end fraction
60. f ( x ) = 4 − 2 x f , open x close , equals 4 minus 2 square root of x
61. f ( x ) = 3 x f , open x close , equals , fraction 3 , over square root of x end fraction
62. f ( x ) = 1 − 2 x f , open x close , equals . fraction 1 , over square root of negative 2 x end root end fraction

63. 1. Open-Ended Copy the mapping diagram below. Complete it by writing members of the domain and range and connecting them with arrows so that r is a function and r − 1 r super negative 1 end super  is not a function.

       

    2. Repeat part (a) so that r is not a function and r − 1 r super negative 1 end super  is a function.
64. Reasoning Relation r has one element in its domain and two elements in its range. Is r a function? Is the inverse of r a function? Explain.
65. Geometry Write a function that gives the length of the hypotenuse of an isosceles right triangle with side length s. Evaluate the inverse of the function to find the side length of an isosceles right triangle with a hypotenuse of 6 in.
66. Open-Ended Write a function f such that the graph of f − 1 f super negative 1 end super  lies only in Quadrants III and IV.
67. Reasoning To determine if the inverse of function f is also a function, you can use a horizontal line test. It says that if no horizontal line intersects the graph of the function f in more than one point, then the inverse of f is a function.
    1. Explain why the horizontal line test works.
    2. The graph of a polynomial function passes through the points ( − 2 , 1 ) , negative 2 comma 1 close comma  (0, 4) and (2, 3). Can its inverse be a function?

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