ESSENTIAL CALCULUS CH05 Inverse functions

1. ESSENTIAL CALCULUSCH05 Inverse functions

2. In this Chapter: • 5.1 Inverse Functions • 5.2 The Natural Logarithmic Function • 5.3 The Natural Exponential Function • 5.4 General Logarithmic and Exponential Functions • 5.5 Exponential Growth and Decay • 5.6 Inverse Trigonometric Functions • 5.7 Hyperbolic Functions • 5.8 Indeterminate Forms and 1’Hospital’s Rule Review

3. Chapter 5, 5.1, P246

4. Chapter 5, 5.1, P246

5. ▓In the language of inputs and outputs, Definition 1 says that f is one-to-one if each output corresponds to only one input. FIGURE 2 This function is not one-to-one because f(x1)=f(x2). Chapter 5, 5.1, P247

6. FIGURE 3 ƒ=(x)=x2 is one-to-one. Chapter 5, 5.1, P247

7. FIGURE 4 g(x)=x2 is not one-to-one. Chapter 5, 5.1, P247

8. 1. DEFINITION A function f is called a one-to-one function if it never takes on the same value twice; that is, F(x1)≠f(x2) whenever x1≠x2 Chapter 5, 5.1, P247

9. HORIZONTAL LINE TEST A function is one-to-one if and only if no horizontal line intersects its graph more than once. Chapter 5, 5.1, P247

10. FIGURE 6 The inverse function reverses inputs and outputs. Chapter 5, 5.1, P248

11. domain of f-1=range of f range of f-1=domain of f Chapter 5, 5.1, P248

12. Do not mistake the -1 in f-1 for an exponent. Thus f-1(x) does not mean Chapter 5, 5.1, P248

13. Chapter 5, 5.1, P248

14. f-1(f(x))=x for every x in A f(f-1(x))=x for every x in B Chapter 5, 5.1, P248

15. Chapter 5, 5.1, P249

16. 5.HOW TO FIND THE INVERSE FUNCTION OF A ONE-TO-ONE FUNCTION f STEP 1 Write =f(x). STEP 2 Solve this equation for x in terms of y (if possible). STEP 3 To express f-1 as a function of x, interchange x and y. The resulting equation is y=f-1(x). Chapter 5, 5.1, P249

17. Chapter 5, 5.1, P250

18. Chapter 5, 5.1, P250

19. The graph of f-1 is obtained by reflecting the graph of f about the line y=x. Chapter 5, 5.1, P250

20. Chapter 5, 5.1, P250

21. 6.THEOREM If f is a one-to-one continuous function defined on an interval, then its inverse function f-1 is also continuous. Chapter 5, 5.1, P251

22. Chapter 5, 5.1, P251

23. 7. THEOREM If f is a one-to-one differentiable function with inverse function f-1 and f’(f-1(a))≠0, then the inverse function is differentiable at a and Chapter 5, 5.1, P251

24. Chapter 5, 5.1, P253

25. 18. The graph of f is given. (a) Why is f one-to-one? (b) What are the domain and range of f-1? (c) What is the value of f-1(2)? (d) Estimate the value of f-1(0) . Chapter 5, 5.1, P253

26. 29–30 ■ Use the given graph of f to sketch the graph of f-1. Chapter 5, 5.1, P253

27. Chapter 5, 5.2, P254

28. Chapter 5, 5.2, P254

29. Chapter 5, 5.2, P254

30. 1.DEFINITION The natural logarithmic function is the function defined by ln x>0 Chapter 5, 5.2, P254

31. Chapter 5, 5.2, P255

32. 3.LAWS OF LOGARITHMS If x and y are positive numbers and r is a rational number, then 1. ln(xy)=ln x+ ln y 2. ln( )=ln x-ln y 3. ln (xr)=r ln x Chapter 5, 5.2, P255

33. Chapter 5, 5.2, P256

34. (a) ln x=∞ (b) ln x=-∞ Chapter 5, 5.2, P256

35. 5. DEFINITION eis the number such that . ln e=1. Chapter 5, 5.2, P257

36. Chapter 5, 5.2, P257

37. Chapter 5, 5.2, P257

38. Chapter 5, 5.2, P258

39. Chapter 5, 5.2, P259

40. Chapter 5, 5.2, P260

41. STEPS IN LOGARITHMIC IFFERENTIATION • Take natural logarithms of both sides of an • equation y=f(x) and use the Laws of Logarithms to simplify. • 2. Differentiate implicitly with respect to x. • 3. Solve the resulting equation for y’. Chapter 5, 5.2, P260

42. Chapter 5, 5.3, P262

43. Chapter 5, 5.3, P262

44. and Chapter 5, 5.3, P262

45. Chapter 5, 5.3, P263

46. Chapter 5, 5.3, P263

47. x>0 Chapter 5, 5.3, P263

48. for all x Chapter 5, 5.3, P263

49. Chapter 5, 5.3, P264

50. 6. PROPERTIES OF THE NATURAL EXPONENTIAL FUNCTION The exponential function f(x)=ex is an increasing continuous function with domain R and range(0,∞) . Thus, ex>0 for all x. Also So the x-axis is a horizontal asymptote of f(x)=ex Chapter 5, 5.3, P264