Chapter 5

1. The Exponential and Logarithmic Functions Chapter 5

2. Inverse Functions Section 5.1

3. Motivation Recall that a function has each domain value x corresponding to exactly one range value y. Some functions have the additional property that each range element y corresponds to exactly one domain value x as well. These are called one-to-one functions.

4. Definition • A function f is a one-to-one function if and only if for each range value, there corresponds exactly one domain value.

5. One-to-One Functions: Equivalent Forms The one-to-one property of a function f can also be stated algebraically in the ff. equivalent forms: 1. If x1≠ x2, then f(x1) ≠ f(x2). 2. If f(x1) = f(x2), then x1 = x2.

6. Example 1 • Prove algebraically that is a one-to-one function.

7. Example 2 • Show that the function is not one-to-one in the interval -5  x  5.

8. Horizontal Line Test for 1-1 Functions A function f is one-to-one if and only if every horizontal line intersects the graph of the function in no more than one point.

9. Theorem • If a function f is increasing throughout its domain or decreasing throughout its domain, then f is a one-to-one function.

10. Motivation • Suppose that we start with the one-to-one function y=x3 and interchange variables to obtain x=y3. Solving for y yields . • Letting f(x) = x3 and g(x) = , note that (f o g)(x) = x and (g o f)(x) = x. • This shows us an example of inverse functions.

11. Definition Two functions f and g are said to be inverse functions if and only if 1. For each x in the domain of g, g(x) is in the domain of f, and (f o g)(x) = f(g(x)) = x. 2. For each x in the domain of f, f(x) is in the domain of g, and (g o f)(x) = g(f(x)) = x.

12. Remarks • The notation f-1 is often used to represent the inverse of f. • In the expression f-1(x), -1 is not an exponent. That is, However,

13. Illustration • f(x) = x3 and are inverse functions since • The following formulas which convert degrees celsius to degrees fahrenheit are inverses:

14. Example • Consider the function , x0. Show by contradiction that f has no inverse.

15. Theorem* • A function f has an inverse f-1 if and only if f is one-to-one.

16. Finding the Inverse of a Function Verify that f is a one-to-one on its domain. Solve the equation y = f(x) for x in terms of y, obtaining an equation of the form x = f(y). Interchange x and y in the previous step. This expresses f-1 as a function of x.

17. Example 1 • Find the inverse f-1 for the function

18. Example 2 • Let f be the function . Find f-1 and sketch the graphs of f and f-1 on the same set of coordinate axes.

19. Remarks It can be shown that if f and g are inverses, then their graphs are reflectionsof each other with respect to the line y=x. Domain of f = Range of f-1 Range of f = Domain of f-1 If f is not one-to-one, it may be possible to restrict the domain of f so that it will have an inverse in that interval(s).

20. Example Obtain two one-to-one functions f1(x) and f2(x) from f(x) = x2 – x by restricting the domain of f. Then find the inverses of these one-to-one functions.

21. The Exponential Function Section 5.2

22. Definition Let a be a positive number not equal to 1. The function defined by the equation y=ax for all real numbers x is called an exponential function with base a.

23. Graph of the Exponential Function Depending on the value of a, the exponential function y=ax has the ff. graph: a>1 0<a<1

24. Properties of the Exponential Function • The domain consists all real numbers. • The range consists of all positive real numbers. That is, ax > 0 for all real numbers x. • The function is one-to-one on R. That is, if and only if x1 = x2 . • The function is increasing when a > 1 and decreasing when 0 < a < 1.

25. Properties of the Expo. Function The y-intercept of the graph is 1. There is no x-intercept. The x-axis is a horizontal asymptote of the graph. There are no vertical asymptotes.

26. Example 1 • Sketch the graph of the function and determine its domain and range.

27. Example 2 • Find an exponential function of the form f(x) = bax + c satisfying the following conditions: • The x-intercept is 1; • The horizontal asymptote is at y = -15; and • The graph passes through the point (2,30).

28. Exponential Equations To solve equations involving exponential functions, it is useful to keep in mind the laws of exponents and the properties of exponential functions. Also, as much as possible, express both sides of the equation as powers of the same base.

29. Examples Solve for x in the ff. equations: (a) (b)

30. Compound Interest • An investment earning compound interest means that the interest earned after each period of time is added to the principal and will itself earn interest in the next time period.

31. Illustration • If a sum of money P, the principal, earns interest at a yearly rate of r, with the interest compounding annually, then After 1 year: P(1+r) After 2 years: P(1+r)2 After 3 years: P(1+r)3 etc. After t years: P(1+r)t

32. Simple Compound Interest Formula where P = principal r = annual interest rate (in decimal) n = no. of interest periods per year t = no. of years P is invested A = final value after t years

33. Example • An investment of P25,000 is placed in a time deposit account that pays an annual interest rate of 5.15%. If no withdrawals or additional deposits are made, how much will there be in the account after 3½ years if interest is compounded (a) yearly? (d) monthly? (b) semi-annually? (e) daily? (c) quarterly?

34. Remark • In the previous example, we see that the more often interest is compounded in a year, the higher the return of investment. • If we let n increase without bound, we have continuously compound interest.

35. Continuous Compound Interest Formula where P = principal r = annual interest rate (in decimal) t = number of years P is invested A = final amount after t years

36. Example • Find the value of a $1000 investment compounded continuously at an annual rate of 8% for 10 years.

37. Behavior of (1+1/k)k

38. Definition The natural exponential function is the function defined by the equation y=ex for all real numbers x, where e is an irrational number whose value is approximately 2.71828.

39. The Logarithmic Function Section 5.3

40. Definition If y=ax where a>0 and a≠1, then the exponent of x of a that produces y is called the logarithm of y to the base a. In symbols, x = logay. That is, x = logay  y = ax

41. Examples Since 10-3 = 1/1000, then log10(1/1000) = -3. log3(1/81) = 4 since 3-4 = 1/81. Since 84/3 = 16, we have log816 = 4/3.

42. Example Evaluate the ff. expression:

43. Remark • Whenever the base is not indicated in a logarithmic expression, it is assumed to be equal to 10.

44. Examples • Solve for x: (a) (b) (c)

45. Definition For any constant a>0, a≠1, the equation y=logax defines a logarithmic function with base a. It is the inverse of the exponential function defined by y=ax.

46. Remarks Since the exponential and logarithmic functions are inverses, by the properties of inverse functions, for all x>0 for all xR The domain of the logarithmic function is all positive real numbers and its range is all real numbers.

47. Example 1 Evaluate the following: (a) (b)

48. Example 2 • Find the domain of the following functions: (a) f(x) = log1/2[-(x – 3)] + 1 (b) g(x) = log x – log (x – 1)

49. Graph of the Logarithmic Function y=logax, a>1 y=logax, 0<a<1

50. Properties of the Log. Function The domain is consists of all positive real numbers. The range consists of all real numbers. The function is one-to-one. The function is increasing when a>1, and decreasing when 0<a<1.