374 CHAPTER 5 Exponential and Logarithmic Functions 20. Find the average rate of change of ( ) = f x 8x from 1 3 to 2 3 . 21. Use the Intermediate Value Theorem to show that ( ) = − + − + f x x x x 2 5 1 4 3 has a zero in the interval [ ] − − 2, 1 . Then, approximate the zero correct to two decimal places. 16. Find the function whose graph is the shape of = y x, but shifted to the right 4 units and reflected about the x-axis. 17. Solve: − − = x x 3 4 5 0 2 18. Solve: + − ≥ x x 1 25 0 2 19. Use the Remainder Theorem to find the remainder when ( ) = − − + − f x x x x x 3 7 27 67 36 5 4 3 2 is divided by +x 3. Is +x 3 a factor of ( ) f x ? Chapter Review Things to Know Composite function (p. 273) ( ) ( ) ( ) ( ) = f g x f g x ; The domain of f g is the set of all numbers x in the domain of g for which ( ) g x is in the domain of f. One-to-one function f (p. 281) A function for which any two different inputs in the domain correspond to two different outputs in the range For any choice of elements x x , 1 2 in the domain of f, if ≠ x x , 1 2 then ( ) ( ) ≠ f x f x . 1 2 Horizontal-line test (p. 282) If every horizontal line intersects the graph of a function f in at most one point, f is one-to-one. Inverse function −f 1 of f (pp. 283, 285, 286) For a one-to-one function ( ) = y f x , the correspondence from the range of f to the domain of f. Domain of = − f f range of ;1 range of = − f f domain of 1 ( ) ( ) = −f f x x 1 for all x in the domain of f ( ) ( ) = − f f x x 1 for all x in the domain of −f 1 The graphs of f and −f 1 are symmetric with respect to the line = y x. Properties of the exponential function (pp. 296, 300, 301) ( ) = > > f x Ca a C , 1, 0 x Domain: the interval ( ) −∞ ∞, Range: the interval ( )∞ 0, x-intercepts: none; y-intercept: C Horizontal asymptote: x-axis ( ) = y 0 as →−∞ x Increasing; one-to-one; smooth; continuous See Figure 23 for a typical graph. ( ) = < < > f x Ca a C , 0 1, 0 x Domain: the interval ( ) −∞ ∞, Range: the interval ( )∞ 0, x-intercepts: none; y-intercept: C Horizontal asymptote: x-axis ( ) = y 0 as →∞ x Decreasing; one-to-one; smooth; continuous See Figure 28 for a typical graph. Number e (p. 302) Number approached by the expression ( ) + n 1 1 n as →∞ n Property of exponents (p. 304) If = a a , u v then = u v. Natural logarithm (p. 316) = y x ln if and only if = x e .y Properties of the logarithmic function (p. 321) ( ) = > f x x a log , 1 a ( ) = = y x x a log if and only if a y Domain: the interval ( )∞ 0, Range: the interval ( ) −∞ ∞, x-intercept: 1; y-intercept: none Vertical asymptote: = x 0 (y-axis) Increasing; one-to-one; smooth; continuous See Figure 44(a) for a typical graph.
RkJQdWJsaXNoZXIy NjM5ODQ=