312 CHAPTER 5 Exponential and Logarithmic Functions 134. Historical Problem Pierre de Fermat (1601–1665) conjectured that the function ( ) = + ( ) f x 2 1 2x for = x 1, 2, 3, . . . , would always have a value equal to a prime number. But Leonhard Euler (1707–1783) showed that this formula fails for = x 5. Use a calculator to determine the prime numbers produced by f for = x 1, 2, 3, 4. Then show that ( ) = × f 5 641 6,700,417, which is not prime. 135. Challenge Problem Solve: 3 4 3 9 0 x x 2 1 − ⋅ + = − 136. Challenge Problem Solve: − ⋅ − = + 2 3 2 20 0 x x 2 3 1 1 3 Problems 132 and 133 define two other transcendental functions. 132. The hyperbolic sine function, designated by x sinh , is defined as ( ) = − − x e e sinh 1 2 x x (a) Show that ( ) = f x x sinh is an odd function. (b) Graph ( ) = f x x sinh . 133. The hyperbolic cosine function, designated by x cosh , is defined as ( ) = + − x e e cosh 1 2 x x (a) Show that ( ) = f x x cosh is an even function. (b) Graph ( ) = f x x cosh . (c) Refer to Problem 132. Show that, for every x, ( ) ( ) − = x x cosh sinh 1 2 2 Explaining Concepts 137. The bacteria in a 4-liter container double every minute.After 60 minutes the container is full. How long did it take to fill half the container? 138. Explain in your own words what the number e is. Provide at least two applications that use this number. 139. Do you think that there is a power function that increases more rapidly than an exponential function? Explain. 140. As the base a of an exponential function ( ) = f x a ,x where > a 1, increases, what happens to its graph for > x 0? What happens to its graph for < x 0? 141. The graphs of = − y a x and ( ) = y a 1 x are identical. Why? Problems 142–151 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 142. Solve the inequality: + ≤ + x x x 5 4 20. 3 2 143. Solve the inequality: + − ≥ x x 1 2 1. 144. Find the equation of the quadratic function f that has its vertex at ( ) 3, 5 and contains the point ( ) 2, 3 . 145. Suppose ( ) = + − f x x x2 3. 2 (a) Graph f by determining whether its graph is concave up or concave down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. (b) Find the domain and range of f. (c) Determine where f is increasing and where it is decreasing. 146. Solve: ( ) ( ) − − = − − x x x x 13 5 6 2 8 27 147. Find an equation for the circle with center ( ) 0, 0 and radius = r 1. 148. Solve: − + = x x 16 48 0 149. If $12,000 is invested at 3.5% simple interest for 2.5 years, how much interest is earned? 150. Determine where the graph of ( ) = − − f x x x5 6 4 2 is below the graph of ( ) = + g x x2 12 2 by solving the inequality ( ) ( ) ≤ f x g x . 151. Find the difference quotient of ( ) = − f x x x 2 7 . 2 ‘Are You Prepared?’ Answers 1. 64; 4; 1 9 2. { } −4, 1 3. False 4. 3 5. True 6. line; −3; 10 7. [ ) ( ] ∞ −∞ 2, ; , 2
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