Chapter Review 377 8. The graph of a function f is given below. State why f is one-to-one. Then draw the graph of the inverse function −f .1 x y 4 –4 4 –4 (3, 3) y = x (2, 0) (0, –2) (–1, –3) In Problems 9 and 10, verify that the functions f and g are inverses of each other by showing that ( ) ( ) = f g x x and ( ) ( ) = g f x x. Give any values of x that need to be excluded from the domain of f and the domain of g. 9. ( ) ( ) = − = + f x x g x x 5 10; 1 5 2 10. ( ) ( ) = − = − f x x x g x x 4 ; 4 1 In Problems 11–14, each function is one-to-one. Find the inverse of each function and check your answer. Find the domain and range of f and −f 1. 11. ( ) = + − f x x x 2 3 5 2 12. ( ) = − f x x 1 1 13. ( ) = − f x x 2 14. ( ) = + f x x 1 1/3 15. Given ( ) = f x 3x and ( ) = g x x log , 3 evaluate each of the following. (a) ( ) f 4 (b) ( ) g 9 (c) ( ) − f 2 (d) ( ) g 1 27 16. Change = z 52 to an equivalent statement involving an logarithm. 17. Change = u log 13 5 to an equivalent statement involving an exponent. In Problems 18 and 19, find the domain of each logarithmic function. 18. ( ) ( ) = − f x x log 3 2 19. ( ) ( ) = − + H x x x log 3 2 2 2 In Problems 20–22, find the exact value of each expression. Do not use a calculator. 20. ( ) log 1 8 2 21. e ln 2 22. 2log 0.4 2 In Problems 23–26, write each expression as the sum and/or difference of logarithms. Express powers as factors. 23. ( ) > > > uv w u v w log 0, 0, 0 3 2 24. ( ) > > a b a b log 0, 0 2 2 4 25. ( ) + > x x x log 1 0 2 3 26. ( ) + − + > x x x x ln 2 3 3 2 2 2 2 In Problems 27–29, write each expression as a single logarithm. 27. + x x 3 log 1 2 log 4 2 4 28. ( ) ( ) ( ) − + + − − x x x x x ln 1 ln 1 ln 1 2 29. ( ) ( ) [ ] + − − − + x x x 1 2 ln 1 4 ln 1 2 1 2 ln 4 ln 2 30. Use the Change-of-Base Formula and a calculator to evaluate log 19. 4 Round your answer to three decimal places. 31. Graph = y x log3 using a graphing utility and the Change-of-Base Formula. In Problems 32–35, for each function f: (a) Find the domain of f. (b) Graph f. (c) From the graph, determine the range and any asymptotes of f. (d) Find −f ,1 the inverse function of f. (e) Find the domain and the range of −f .1 (f) Graph −f .1 32. ( ) = − f x 2x 3 33. ( ) = + − f x 1 3 x 34. ( ) = − f x e3 x 2 35. ( ) ( ) = + f x x 1 2 ln 3
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