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Unformatted text preview: EXERCISES 7.1
Determine whether or not the given function is onetoone and, if so, find the inverse. If f has an inverse, give the domain of f 1 . 1. f (x) = 5x + 3. 3. f (x) = 1  x .
2 17. f (x) = sin x, 18. f (x) = cos x, 19. f (x) = 1 . x  x . 2 2  x . 2 2 20. f (x) = 1 . 1x x 22. f (x) = . x 24. f (x) = 2. f (x) = 3x + 5. 4. f (x) = x5 . 6. f (x) = x2  3x + 2. 8. f (x) = x3  1. 10. f (x) = (1  x)4 . 12. f (x) = (4x  1)3 . 14. f (x) = 1  (x  2)1/3 . 16. f (x) = (2  3x2 )3 . 5. f (x) = x5 + 1. 7. f (x) = 1 + 3x3 . 9. f (x) = (1  x)3 . 11. f (x) = (x + 1)3 + 2. 13. f (x) = x3/5 . 15. f (x) = (2  3x)3 . 1 21. f (x) = x + . x 1 . +1 x+2 25. f (x) = . x+1 23. f (x) = x3 1  1. 1x 1 26. f (x) = . (x + 1)2/3 27. What relation is there between f and ( f 1 )1 ? 7.1 ONETOONE FUNCTIONS; INVERSES 379 In Exercises 2831, sketch the graph of f 1 given the graph of f . 28.
y In Exercise 4547, find a formula for ( f 1 ) (x) given that f is onetoone and its derivative satisfies the indicated equation. 45. f (x) = f (x). 46. f (x) = 1 + [ f (x)]2 . 47. f (x) = 48. Let 1  [ f (x)]2 . f (x) = x3  1, x < 0. x2 , x 0. 29.
y 1 1 1 x 1 1 x (a) Sketch the graph of f and verify that f is onetoone. (b) Find f 1 . In Exercises 49 and 50, let f (x) = ax + b , cx + d d x= . c 30.
y 31.
y 49. (a) Show that f is onetoone iff ad  bc = 0. (b) Suppose that ad  bc = 0. Find f 1 . 50. Determine the constants a, b, c, d so that f = f 1 . 51. Let
x 1 1 x 1 1 x f (x) =
2 1 + t 2 dt. (a) Prove that f has an inverse. (b) Find ( f 1 ) (0). 52. Let
2x 32. (a) Show that the composition of two onetoone functions, f and g, is onetoone. (b) Express ( f g)1 in terms of f 1 and g 1 . 33. a. Let f (x) = 1 x3 + x2 + kx, k a constant. For what values 3 of k is f onetoone? b. Let g(x) = x3 + kx2 + x, k a constant. For what values of k is g onetoone? 34. a. Suppose that f has an inverse and f (2) = 5, f (2) =  3 . 4 What is ( f 1 ) (5)? b. Suppose that f has an inverse and f (2) = 3, f (2) = 2 . 3 If g = 1/f 1 , what is g ( 3)? In Exercises 3544, the given function f is differentiable. Verify that f has an inverse and find ( f 1 ) (c). 35. f (x) = x3 + 1; c = 9. 36. f (x) = 1  2x  x3 ; c = 4. 37. f (x) = x + 2 x, x > 0; c = 8. 38. f (x) = sin x,  < x < ; 2 2 c = 1. 2 39. f (x) = 2x + cos x; c = . x+3 40. f (x) = , x > 1; c = 3. x1 41. f (x) = tan x,  < x < ; 2 2 c= f (x) =
1 16 + t 4 dt. (a) Prove that f has an inverse. (b) Find ( f 1 ) (0). x 53. Let f be defined by f (x) = 3 g(t) dt, where g is a function continuous on [a, b]. (a) What conditions on g will imply that f has an inverse? (b) Given that f has an inverse, what conditions on g will imply that f is differentiable? (c) Given that f 1 is differentiable, find ( f 1 ) . 54. Suppose that the function f has an inverse and fix a number c = 0. (a) Let g(x) = f (x + c) for all x such that x + c dom ( f ). Prove that g has an inverse and find it. (b) Let h(x) = f (cx) for all x such that cx dom ( f ). Prove that h has an inverse and find it. 55. Let f be a onetoone, twice differentiable function and let g = f 1 . (a) Show that g (x) =  f [g(x)] . ( f [g(x)]3 ) 3. 42. f (x) = x5 + 2x3 + 2x; c = 5. 1 43. f (x) = 3x  3 ; x > 0, c = 2. x 44. f (x) = x  + cos x, 0 < x < 2, c = 1. (b) Suppose that the graph of f is concave up (down) on an interval I . What can you say about the graph of f 1 ? 56. Let P(x) = an xn +an1 xn1 + +a1 x+a0 be a polynomial. (a) Can P have an inverse if its degree n is even? Justify your answer. 380 CHAPTER 7 THE TRANSCENDENTAL FUNCTIONS (b) Can P have an inverse if n is odd? If so, give an example. Also, give an example of a polynomial of odd degree that does not have an inverse. 57. Suppose that f is differentiable and has an inverse. If we let y = f 1 (x), then f (y) = x and we can find dy/dx by differentiating implicitly. The function f (x) = sin x,  < x < , 2 2 is onetoone and differentiable. Let y = f 1 (x) and find dy/dx. Express your result as a function of x. 58. Repeat Exercise 57 for the function f (x) = tan x,  < x < . 2 2 In Exercises 5962, use a CAS to find f 1 . Verify your result by showing that f [ f 1 (x)] = x and f 1 [ f (x)] = x. 59. f (x) = 2  (x + 1)3 . 3x 60. f (x) = , x = 5. 2 2x + 5 61. f (x) = 4 + 3 x  1, x 1. 62. f (x) = 3 8  x + 2. Use a CAS in Exercises 6364 to find f 1 and then show that 1 . ( f 1 ) (x) = f [ f 1 (x)] 1x . 1+x 64. f (x) = 1  3x + 5, 63. f (x) = x 5. 3 c In Exercises 6568, use a graphing utility to draw the graph of f and verify that f is onetoone. Then draw the graph of f 1 ; graph f and f 1 together.
65. f (x) = x3 + 3x + 2. 67. f (x) = 4 sin 2x, 68. f (x) = 2  cos 3x, 66. f (x) = x3/5  1.  x . 4 4 0 x . 3 ...
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 Spring '10
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