SECTION 2.4 Library of Functions; Piecewise-defined Functions 111 8 ft 8 ft 8 ft 8 ft 3 ft 6 ft 3 ft 16 ft 65. Challenge Problem Find the sum function f g x ( )( ) + if ( ) = + < + ≥ ⎧ ⎨ ⎪⎪ ⎩⎪⎪ f x x x x x x 2 3 if 2 5 if 2 2 and g x x x x x 4 1 if 0 7 if 0 ( ) = − + ≤ − > ⎧ ⎨ ⎪⎪ ⎩⎪⎪ 62. Wind Chill Redo Problem 61(a)–(d) for an air temperature of 10C. − ° 63. Sales Commission A salesperson earns $45,000 per year plus 4% commission on all sales up to (and including) $250,000. If sales exceed $250,000 in a calendar year, the salesperson’s commission jumps to 6% for all their sales over $250,000 up to (and including) $500,000. For sales of more than $500,000, the salesperson’s commission jumps to 9% of such sales. Develop a model that expresses S, the total salary for a given year, as a function of x, the total sales in dollars. 64. Challenge Problem Pool Depth Develop a model for the depth of the swimming pool shown below as a function of the distance from the wall on the left. Explaining Concepts In Problems 66–73, use a graphing utility. 66. Exploration Graph y x .2 = Then on the same screen graph y x 2, 2 = + followed by y x 4, 2 = + followed by y x 2. 2 = − What pattern do you observe? Can you predict the graph of y x 4? 2 = − Of y x 5? 2 = + 67. Exploration Graph y x .2 = Then on the same screen graph y x 2 , 2 ( ) = − followed by y x 4 , 2 ( ) = − followed by y x 2 . 2 ( ) = + What pattern do you observe? Can you predict the graph of y x 4 ?2 ( ) = + Of y x 5 ?2 ( ) = − 68. Exploration Graph y x . = Then on the same screen graph y x 2 , = followed by y x 4 , = followed by y x 1 2 . = What pattern do you observe? Can you predict the graph of y x 1 4 ? = Of y x 5 ? = 69. Exploration Graph y x .2 = Then on the same screen graph y x .2 = − Now try y x = and y x . = − What do you conclude? 70. Exploration Graph y x. = Then on the same screen graph y x. = − Now try y x2 1 = + and y x 2 1. ( ) = − + What do you conclude? 71. Exploration Graph y x .3 = Then on the same screen graph y x 1 2. 3 ( ) = − + Could you have predicted the result? 72. Exploration Graph y x y x , , 2 4 = = and y x6 = on the same screen. What do you notice is the same about each graph? What do you notice is different? 73. Exploration Graph y x y x , , 3 5 = = and y x7 = on the same screen. What do you notice is the same about each graph? What do you notice is different? 74. Consider the equation y x x 1 if is rational 0 if is irrational = ⎧ ⎨ ⎪ ⎩⎪⎪ Is this a function? What is its domain? What is its range? What is its y-intercept, if any? What are its x-intercepts, if any? Is it even, odd, or neither? How would you describe its graph? 75. Define some functions that pass through 0, 0 ( ) and 1, 1 ( ) and are increasing for x 0. ≥ Begin your list with y x y x , , = = and y x .2 = Can you propose a general result about such functions? Retain Your Knowledge Problems 76–85 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. 76. Simplify: x y3 5 2 ( ) − − 77. Find the center and radius of the circle x y y6 16. 2 2 + = + 78. Solve: x x x 4 5 2 1 4 7 1 ( ) ( ) − − = − + 79. Ethan has $60,000 to invest. He puts part of the money in a CD that earns 3% simple interest per year and the rest in a mutual fund that earns 8% simple interest per year. How much did he invest in each if his earned interest the first year was $3700? 80. Find the quotient and remainder when x x3 6 3 2 + − is divided by x 2. + 81. What is the conjugate of i 3 2 2 ? − [This problem is based on content from Section A.7, which is optional.] 82. Identify the leading term: x x x 5 8 2 4 2 7 − + − 83. Simplify: t t 5 25 2 2 7 2 2 ( ) ( ) + 84. Find the domain of h x x x 7 7 . 4 ( ) = + + 85. Factor: x y x y x y 3 2 18 12 3 2 2 − + − ‘Are You Prepared?’ Answers 1. x y 4 2 (4, 2) (1, 1) (0, 0) 2. (21, 21) x y 2 2 (1, 1) 3. 0, 8, 2,0 ( ) ( ) −
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