SECTION 2.1 Functions 75 59. ( ) = − h x x3 12 60. ( ) = − G x x 1 61. ( ) = + − p x x x2 3 1 62. ( ) = − − − f x x x 1 3 1 4 63. ( ) = − f x x x 4 64. ( ) = − − − f x x x 2 65. ( ) = − − P t t t 4 3 21 66. ( ) = + − h z z z 3 2 67. ( ) = − f x x5 4 3 68. ( ) = − + + g t t t t7 2 3 2 69. ( ) = + − − M t t t t 1 5 14 2 5 70. ( ) = − N p p p2 98 2 5 In Problems 83–98, find the difference quotient of f; that is, find ( ) ( ) + − ≠ f x h f x h h , 0, for each function. Be sure to simplify. 83. ( ) = + f x x4 3 84. ( ) = − + f x x3 1 85. ( ) = − f x x 4 2 86. ( ) = + f x x3 2 2 87. ( ) = − + f x x x 4 2 88. ( ) = − + f x x x 3 2 6 2 89. ( ) = − f x x 5 4 3 90. ( ) = + f x x 1 3 91. ( ) = + f x x x 2 3 92. ( ) = − f x x x 5 4 93. ( ) = − f x x 2 94. ( ) = + f x x 1 95. ( ) = f x x 1 2 96. ( ) = + f x x 1 1 2 97. ( ) = − f x x 4 2 98. ( ) = + f x x 1 2 In Problems 71–80, for the given functions f and g, find the following. For parts (a)–(d), also find the domain. (a) ( )( ) + f g x (b) ( )( ) − f g x (c) ( )( ) ⋅ f g x (d) ( ) ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ f g x (e) ( )( ) + f g 3 (f) ( )( ) − f g 4 (g) ( )( ) ⋅ f g 2 (h) ( ) ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ f g 1 71. ( ) ( ) = + = − f x x g x x 3 4; 2 3 72. ( ) ( ) = + = − f x x g x x 2 1; 3 2 73. ( ) ( ) = − = f x x g x x 1; 2 2 74. ( ) ( ) = + = + f x x g x x 2 3; 4 1 2 3 75. ( ) ( ) = = − f x x g x x ; 3 5 76. ( ) ( ) = = f x x g x x ; 77. ( ) ( ) = + = f x x g x x 1 1 ; 1 78. ( ) ( ) = − = − f x x g x x 1; 4 79. ( ) ( ) = + − = − f x x x g x x x 2 3 3 2 ; 4 3 2 80. ( ) ( ) = + = f x x g x x 1; 2 81. Given ( ) = + f x x3 1 and ( )( ) + = − f g x x 6 1 2 , find the function g. 82. Given ( ) = f x x 1 and ( ) ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ = + − f g x x x x 1 , 2 find the function g. 99. If ( ) = − + f x x x2 3, 2 find the value(s) of x so that ( ) = f x 11. 100. If ( ) = − f x x 5 6 3 4 , find the value(s) of x so that ( ) = − f x 7 16 . 101. If ( ) = + + − f x x Ax x 2 4 5 3 2 and ( ) = f 2 5, what is the value of A? 102. If ( ) = − + f x x Bx 3 4 2 and ( ) − = f 1 12, what is the value of B? 103. If ( ) = + − f x x x A 3 8 2 and ( ) = f 0 2, what is the value of A? 104. If ( ) = − + f x x B x 2 3 4 and ( ) = f 2 1 2 , what is the value of B? 105. Geometry Express the area A of a rectangle as a function of the length x if the length of the rectangle is twice its width. 106. Geometry Express the area A of an isosceles right triangle as a function of the length x of one of the two equal sides. 107. Constructing Functions Express the gross wages G of a person who earns $16 per hour as a function of the number x of hours worked. 108. Constructing Functions Ann, a commissioned salesperson, earns $100 base pay plus $10 per item sold. Express her gross salary G as a function of the number x of items sold. 109. Effect of Gravity on Earth If a rock falls from a height of 20 meters on Earth, the height H (in meters) after x seconds is approximately ( ) = − H x x 20 4.9 2 (a) What is the height of the rock when = x 1 second? When = x 1.1 seconds? When = x 1.2 seconds? (b) When is the height of the rock 15 meters? When is it 10 meters? When is it 5 meters? (c) When does the rock strike the ground? Applications and Extensions
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