294 CHAPTER 5 Exponential and Logarithmic Functions 113. Explain why the horizontal-line test can be used to identify one-to-one functions from a graph. 114. Explain why a function must be one-to-one in order to have an inverse that is a function. Use the function = y x2 to support your explanation. 110. Give an example of a function whose domain is the set of real numbers and that is neither increasing nor decreasing on its domain, but is one-to-one. [ Hint : Use a piecewise-defined function.] 111. Is every odd function one-to-one? Explain. 112. Suppose that C g( ) represents the cost C , in dollars, of manufacturing g cars. Explain what ( ) −C 800,000 1 represents. ‘Are You Prepared?’ Answers 1. x 2. Increasing on [ )∞ 0, ; decreasing on ( ] −∞, 0 3. { } ≠ − ≠ x x x 6, 3 4. − ≠ ≠ − x x x x 1 , 0, 1 Problems 115–124 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 115. If ( ) = − f x x x 3 7 , 2 find ( ) ( ) + − f x h f x . 116. Find the zeros of the quadratic function ( ) = + + f x x x 3 5 1 2 What are the x -intercepts, if any, of the graph of the function? Find the vertex. Is it a maximum or minimum? Is the graph concave up or concave down? 117. Use the techniques of shifting, compressing or stretching, and reflections to graph f x x 2 3. ( ) = − + + 118. Find the domain of ( ) = − − − − R x x x x x 6 11 2 2 6 . 2 2 Find any horizontal, vertical, or oblique asymptotes. 119. Find an equation of a circle with center ( ) −3, 5 and radius 7. 120. Find an equation of the line that contains the point ( ) −4, 1 and is perpendicular to the line − = x y 3 6 5. Write the equation in slope-intercept form. 121. Is the function ( ) = + f x x x x 3 5 7 3 even, odd, or neither? 122. Solve for D : + = + x yD xD y 2 2 123. Find the average rate of change of ( ) = − + + f x x x 3 2 1 2 from 2 to 4. 124. Find the difference quotient of ( ) = + f f x x : 2 3 5.3 Exponential Functions Now Work the ‘Are You Prepared?’ problems on page 306. • Exponents (Section A.1, pp. A7–A9, and Section A.10, pp. A91–A92) • Graphing Techniques: Transformations (Section 2.5, pp. 112–120) • Solving Linear and Quadratic Equations (Section A.6, pp. A45–A54 • Average Rate of Change (Section 2.3, pp. 93–95) • Quadratic Functions (Section 3.3, pp. 157–166) • Linear Functions (Section 3.1, pp. 139–145) • Horizontal Asymptotes (Section 4.5, pp. 241–244) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Evaluate Exponential Functions (p. 294) 2 Graph Exponential Functions (p. 298) 3 Define the Number e (p. 302) 4 Solve Exponential Equations (p. 304) 1 Evaluate Exponential Functions Section A.10 gives a definition for raising a real number a to a rational power. That discussion provides meaning to expressions of the form ar where the base a is a positive real number and the exponent r is a rational number.
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