100 CHAPTER 2 Functions and Their Graphs 104. Show that a constant function f x b ( ) = has an average rate of change of 0. Compute the average rate of change of y x 4 2 = − on the interval 2, 2 . [ ] − Explain how this can happen. 103. A function f has a positive average rate of change on the interval 2, 5 . [ ] Is f increasing on 2, 5 ? [ ] Explain. ‘Are You Prepared?’ Answers 1. < < x 2 5 2. 1 3. symmetric with respect to the y -axis 4. ( ) + = − y x 2 5 3 5. ( ) ( ) ( ) − − 3,0, 3,0, 0, 9 110. Factor completely: x x x x x x 2 2 3 1 15 3 1 3 2 2 2 3 5 4 5 2 2 2 ( ) ( ) ( ) ( ) − ⋅ + ⋅ + + ⋅ − ⋅ 111. Find the midpoint of the line segment connecting the points 2, 1 ( ) − and 3 5 , 4. ( ) − 112. Solve: x3 7 3 5 + − = 113. Find the real solutions of x x7 8. 6 3 + = 114. Solve for D if y D x xy x y D 3 3 3 3 0 2 2 2 2 ⋅ + − − ⋅ = 105. Simplify: 540 106. Multiply: a b 4 2 ( ) − 107. The daily rental charge for a moving truck is $40 plus a mileage charge of $0.80 per mile. Express the cost C to rent a moving truck for one day as a function of the number x of miles driven. 108. The shelf life of a perishable commodity varies inversely with the storage temperature. If the shelf life at 10 C° is 33 days, what is the shelf life at 40 C? ° 109. Write the standard form of the equation of a circle with center 3, 2 ( ) − and radius r 6 2 . = Problems 105–114 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. Retain Your Knowledge OBJECTIVES 1 Graph the Functions Listed in the Library of Functions (p. 100) 2 Analyze a Piecewise-defined Function (p. 105) 2.4 Library of Functions; Piecewise-defined Functions Now Work the ‘Are You Prepared?’ problems on page 108. • Intercepts (Section 1.3, pp. 20–21) • Graphs of Key Equations (Section 1.1: Example 3, p. 5; Examples 7–9, pp. 24–26) PREPARING FOR THIS SECTION Before getting started, review the following: 1 Graph the Functions Listed in the Library of Functions Figure 37 shows the graph of y x. = Based on the vertical-line test, y f x x ( ) = = is a function, called the square root function . From the graph, we have the following properties: Properties of ( ) = f x x • The domain and the range of f x x ( ) = are the set of nonnegative real numbers. • The x -intercept of the graph of f x x ( ) = is 0. • The y -intercept of the graph of f x x ( ) = is also 0. • f x x ( ) = is neither even nor odd. • f x x ( ) = is increasing on the interval 0, . [ )∞ • f x x ( ) = has an absolute minimum of 0 at x 0. = Figure 37 y f x x ( ) = = x y (9, 3) 6 (4, 2) (0, 0) 10 –2 5 (1, 1)
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