292 CHAPTER 5 Exponential and Logarithmic Functions 45. ( ) ( ) = − = + f x x g x x 8; 8 3 3 46. ( ) ( ) ( ) = − ≥ = + f x x x g x x 2 , 2; 2 2 47. ( ) ( ) = = f x x g x x 1 ; 1 48. ( ) ( ) = = f x x g x x ; 49. ( ) ( ) = + + = − − f x x x g x x x 2 3 4 ; 4 3 2 50. ( ) ( ) = − + = + − f x x x g x x x 5 2 3 ; 3 5 1 2 In Problems 51–62, the function f is one-to-one. (a) Find its inverse function −f 1 and check your answer. (b) Find the domain and the range of f and −f .1 (c) Graph f, −f ,1 and = y x on the same coordinate axes. 51. ( ) = f x x3 52. ( ) = − f x x4 53. ( ) = + f x x4 2 54. ( ) = − f x x 1 3 55. ( ) = − f x x 1 3 56. ( ) = + f x x 1 3 57. ( ) = + ≥ f x x x 4, 0 2 58. ( ) = + ≥ f x x x 9, 0 2 59. ( ) = f x x 4 60. f x x 3 ( ) = − 61. ( ) = − f x x 1 2 62. ( ) = + f x x 4 2 81. Use the graph of ( ) = y f x given in Problem 35 to evaluate the following: (a) ( ) − f 1 (b) ( ) f 1 (c) ( ) −f 1 1 (d) ( ) −f 2 1 82. Use the graph of ( ) = y f x given in Problem 36 to evaluate the following: (a) ( ) f 2 (b) ( ) f 1 (c) ( ) −f 0 1 (d) ( ) − −f 1 1 83. If ( ) = f 7 13 and f is one-to-one, what is ( ) −f 13 ? 1 84. If ( ) − = g 5 3 and g is one-to-one, what is ( ) −g 3 ? 1 85. The domain of a one-to-one function f is [ )∞ 5, , and its range is [ ) − ∞ 2, . State the domain and the range of −f .1 86. The domain of a one-to-one function f is [ )∞ 0, , and its range is [ )∞ 5, . State the domain and the range of −f .1 87. The domain of a one-to-one function g is ( ] −∞, 0 , and its range is [ )∞ 0, . State the domain and the range of −g .1 88. The domain of a one-to-one function g is [ ] 0, 15 , and its range is ( ) 0, 8 . State the domain and the range of −g .1 89. A function ( ) = y f x is increasing on the interval 0, 5 . [ ] What conclusions can you draw about the graph of ( ) = − y f x ? 1 Applications and Extensions 90. A function ( ) = y f x is decreasing on the interval 0, 5 . [ ] What conclusions can you draw about the graph of ( ) = − y f x ? 1 91. Find the inverse of the linear function ( ) = + ≠ f x mx b m , 0 92. Find the inverse of the function ( ) = − ≤ ≤ f x r x x r , 0 2 2 93. A function f has an inverse function −f .1 If the graph of f lies in quadrant I, in which quadrant does the graph of −f 1 lie? 94. A function f has an inverse function −f .1 If the graph of f lies in quadrant II, in which quadrant does the graph of −f 1 lie? 95. The function ( ) = f x x is not one-to-one. Find a suitable restriction on the domain of f so that the new function that results is one-to-one.Then find the inverse of the new function. 96. The function ( ) = f x x4 is not one-to-one. Find a suitable restriction on the domain of f so that the new function that results is one-to-one.Then find the inverse of the new function. In Problems 63–80, the function f is one-to-one. (a) Find its inverse function −f 1 and check your answer. (b) Find the domain and the range of f and −f .1 63. ( ) = + f x x 2 3 64. ( ) = − f x x 4 2 65. ( ) = + f x x x 3 2 66. ( ) = − − f x x x 2 1 67. ( ) = − f x x x 2 3 1 68. f x x x 3 1 ( ) = − + 69. ( ) = + − f x x x 3 4 2 3 70. ( ) = − + f x x x 2 3 4 71. ( ) = + + f x x x 2 3 2 72. ( ) = − − − f x x x 3 4 2 73. ( ) = − > f x x x x 4 2 , 0 2 2 74. ( ) = + > f x x x x 3 3 , 0 2 2 75. ( ) = − ≥ f x x x 4, 0 2 3 76. ( ) = + f x x 5 3 2 77. ( ) = − f x x 2 3 5 78. ( ) = + f x x 13 5 3 79. ( ) ( ) = − + ≥ f x x x 1 9 1 2, 1 2 80. ( ) = + − f x x 2 3 5
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