280 CHAPTER 5 Exponential and Logarithmic Functions (c) Find f g and g f. What does each of these functions represent? Which combination of discounts represents a better deal for the consumer? Why? 70. Taxes Suppose that you work for $20 per hour. Write a function that represents gross salary G as a function of hours worked h. Your employer is required to withhold taxes (federal income tax, Social Security, Medicare) from your paycheck. Suppose your employer withholds 20% of your income for taxes. Write a function that represents net salary N as a function of gross salary G. Find and interpret N G. 71. Suppose f x x x x 16 16 3 2 ( ) = + − − and g x x 4. 2 ( ) = − Find the zeros of f g x . ( )( ) 72. Suppose f x x x x 2 3 8 12 3 2 ( ) = − − + and g x x 5. ( ) = + Find the zeros of f g x . ( )( ) 73. Let f x ax b ( ) = + and g x bx a, ( ) = + where a and b are integers. If f 1 8 ( ) = and f g g f 20 20 14, ( ) ( ) ( ) ( ) − = − find the product of a and b.* 74. Challenge Problem If f and g are odd functions, show that the composite function f g is also odd. 75. Challenge Problem If f is an odd function and g is an even function, show that the composite functions f g and g f are both even. 76. Challenge Problem If ( ) ( ) = + + = + fx x xcgx axb 5 , , 2 and f g x x x 4 22 31, 2 ( )( ) = + + find a, b, and c. 77. Challenge Problem Given three functions f, g, and h, define f g h x f g h x . ( ) [ ] ( ) ( ( )) = Find f g h 2 ( )( ) if f x x g x x 6 7, 1 , ( ) ( ) = − = and h x x 7. ( ) = + 78. Challenge Problem If f x x g x x x 1 4 , 2 , ( ) ( ) = + = − and h x x 3, ( ) = + find the domain of f g h x . ( )( ) 65. Volume of a Cylinder The volume V of a right circular cylinder of height h and radius r is π = V r h. 2 If the height is twice the radius, express the volume V as a function of r. 66. Volume of a Cone The volume V of a right circular cone is π = V r h 1 3 . 2 If the height is twice the radius, express the volume V as a function of r. 67. Foreign Exchange Traders often buy foreign currency in the hope of making money when the currency’s value changes. For example, on May 28, 2024, one U.S. dollar could purchase 0.9196 euro, and one euro could purchase 170.622 yen. Let f x( ) represent the number of euros you can buy with x dollars, and let g x( ) represent the number of yen you can buy with x euros. (a) Find a function that relates dollars to euros. (b) Find a function that relates euros to yen. (c) Use the results of parts (a) and (b) to find a function that relates dollars to yen. That is, find g f x . ( )( ) (d) What is g f 1000 ( )( )? 68. Temperature Conversion The function ( ) ( ) = − C F F 5 9 32 converts a temperature in degrees Fahrenheit, F, to a temperature in degrees Celsius, C. The function K C C 273, ( ) = + converts a temperature in degrees Celsius to a temperature in kelvins, K. (a) Find a function that converts a temperature in degrees Fahrenheit to a temperature in kelvins. (b) Determine 80 degrees Fahrenheit in kelvins. 69. Discounts The manufacturer of a computer is offering two discounts on last year’s model computer.The first discount is a $200 rebate and the second discount is 20% off the regular price, p. (a) Write a function f that represents the sale price if only the rebate applies. (b) Write a function g that represents the sale price if only the 20% discount applies. Retain Your Knowledge Problems 79–88 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 79. Given f x x3 8 ( ) = + and g x x 5, ( ) = − find ( ) ( ) ( ) ( ) ( ) ( ) ( ) + − ⋅ ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ f g x f g x f g x f g x , , , and State the domain of each. 80. Find the real zeros of f x x x 2 5 2. ( ) = − + 81. Find the domain of R x x x x 6 5 3 . 2 ( ) = + + − Find any horizontal, vertical, or oblique asymptotes. 82. For the quadratic function ( ) = − + + f x x x 1 3 2 5, 2 find the vertex and the axis of symmetry, and determine whether the graph is concave up or concave down. 83. Solve: x x6 7 0 2 − − ≤ 84. If a right triangle has hypotenuse c 2 = and leg a 1, = find the length of the other leg b. 85. Find the points of intersection of the graphs of the functions f x x x3 7 2 ( ) = + + and g x x2 3. ( ) = − + 86. Find the distance between the points 3, 8 ( ) − and 2, 7. ( ) − 87. Simplify: x c x c 3 1 3 1 + − + − 88. Solve: x x x x 9 2 9 0 3 2 1/2 2 1/2 ( ) ( ) − − + − = − ‘Are You Prepared?’ Answers 1. 21 − 2. x 4 18 2 − 3. { } ≠ − ≠ x x x 5, 5 *Courtesy of the Joliet Junior College Mathematics Department
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