Exponential and Logarithmic Functions

1. Exponential and Logarithmic Functions Chapter 4 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

2. Composite Functions Section 4.1

3. Composite Functions • Construct new function from two given functions f and g • Compositefunction: • Denoted by f °g • Read as “f composed with g” • Defined by (f°g)(x) = f(g(x)) • Domain: The set of all numbers x in the domain of g such that g(x) is in the domain of f.

4. Composite Functions • Note that we perform the inside function g(x) first.

5. Composite Functions

6. Composite Functions • Example. Suppose that f(x) = x3 { 2 and g(x) = 2x2 + 1. Find the values of the following expressions. (a) Problem: (f±g)(1) Answer: (b) Problem: (g±f)(1) Answer: (c) Problem: (f±f)(0) Answer:

7. Composite Functions • Example. Suppose that f(x) = 2x2 + 3 and g(x) = 4x3 + 1. (a) Problem: Find f±g. Answer: (b) Problem: Find the domain of f±g. Answer: (c) Problem: Find g± f. Answer: (d) Problem: Find the domain of f±g. Answer:

8. Composite Functions • Example. Suppose that f(x) = and g(x) = (a) Problem: Find f±g. Answer: (b) Problem: Find the domain of f±g. Answer: (c) Problem: Find g± f. Answer: (d) Problem: Find the domain of f±g. Answer:

9. Composite Functions • Example. Problem: If f(x) = 4x + 2 and g(x) = show that for all x, (f±g)(x) = (g±f)(x) = x

10. Decomposing Composite Functions • Example. Problem: Find functions f and g such that f±g = H if Answer:

11. Key Points • Composite Functions • Decomposing Composite Functions

12. One-to-One Functions;Inverse Functions Section 4.2

13. One-to-One Functions • One-to-one function: Any two different inputs in the domain correspond to two different outputs in the range. • If x1 and x2 are two different inputs of a function f, then f(x1) f(x2).

14. One-to-One Functions • One-to-one function • Not a one-to-one function • Not a function

15. One-to-One Functions • Example. Problem: Is this function one-to-one? Answer: Person Salary Melissa John Jennifer Patrick $45,000 $40,000 $50,000

16. One-to-One Functions • Example. Problem: Is this function one-to-one? Answer: Person ID Number 1451678 1672969 2004783 1914935 Alex Kim Dana Pat

17. One-to-One Functions • Example. Determine whether the following functions are one-to-one. (a) Problem:f(x) = x2 + 2 Answer: (b) Problem:g(x) = x3 { 5 Answer:

18. One-to-One Functions • Theorem. A function that is increasing on an interval I is a one-to-one function on I. A function that is decreasing on an interval I is a one-to-one function on I.

19. Horizontal-line Test • If every horizontal line intersects the graph of a function f in at most one point, then f is one-to-one.

20. Horizontal-line Test • Example. Problem: Use the graph to determine whether the function is one-to-one. Answer:

21. Horizontal-line Test • Example. Problem: Use the graph to determine whether the function is one-to-one. Answer:

22. Inverse Functions • Requires f to be a one-to-one function • The inverse function of f • Written f{1 • Defined as the function which takes • f(x) as input • Returns the output x. • In other words, f{1 undoes the action of f • f{1(f(x)) = x for all x in the domain of f • f(f{1(x)) = x for all x in the domain of f{1

23. Inverse Functions • Example. Find the inverse of the function shown. Problem: Person ID Number 1451678 1672969 2004783 1914935 Alex Kim Dana Pat

24. Inverse Functions • Example. (cont.) Answer: ID Number Person 1451678 1672969 2004783 1914935 Alex Kim Dana Pat

25. Inverse Functions • Example. Problem: Find the inverse of the function shown. f(0, 0), (1, 1), (2, 4), (3, 9), (4, 16)g Answer:

26. Domain and Range of Inverse Functions • If f is one-to-one, its inverse is a function. • The domain of f{1 is the range of f. • The range of f{1 is the domain of f

27. Domain and Range of Inverse Functions • Example. Problem: Verify that the inverse of f(x) = 3x { 1 is

28. Graphs of Inverse Functions • The graph of a function f and its inverse f{1 are symmetric with respect to the line y = x.

29. Graphs of Inverse Functions • Example. Problem: Find the graph of the inverse function Answer:

30. Finding Inverse Functions • If y = f(x), • Inverse if given implicitly by x = f(y). • Solve for y if possible to get y = f {1(x) • Process • Step 1: Interchange x and y to obtain an equation x = f(y) • Step 2: If possible, solve for y in terms of x. • Step 3: Check the result.

31. Finding Inverse Functions • Example. Problem: Find the inverse of the function Answer:

32. Restricting the Domain • If a function is not one-to-one, we can often restrict its domain so that the new function is one-to-one.

33. Restricting the Domain • Example. Problem: Find the inverse of if the domain of f is x¸ 0. Answer:

34. Key Points • One-to-One Functions • Horizontal-line Test • Inverse Functions • Domain and Range of Inverse Functions • Graphs of Inverse Functions • Finding Inverse Functions • Restricting the Domain

35. Exponential Functions Section 4.3

36. Exponents • For negative exponents: • For fractional exponents:

37. Exponents • Example. Problem: Approximate 3¼ to five decimal places. Answer:

38. Laws of Exponents • Theorem. [Laws of Exponents]If s, t, a and b are real numbers with a> 0 and b> 0, then • as¢at = as+t • (as)t = ast • (ab)s = as¢bs • 1s = 1 • a0 = 1

39. Exponential Functions • Exponential function: function of the form f(x) = ax • where a is a positive real number (a> 0) • a 1. • Domain of f: Set of all real numbers. Warning! This is not the same as a power function. (A function of the form f(x) = xn)

40. Exponential Functions • Theorem. For an exponential function f(x) = ax, a > 0, a 1, if x is any real number, then

41. Graphing Exponential Functions • Example. Problem: Graph f(x) = 3x Answer:

42. Graphing Exponential Functions

43. Properties of the Exponential Function • Properties of f(x) = ax, a > 1 • Domain: All real numbers • Range: Positive real numbers; (0, 1) • Intercepts: • No x-intercepts • y-intercept of y = 1 • x-axis is horizontal asymptote as x  {1 • Increasing and one-to-one. • Smooth and continuous • Contains points (0,1), (1, a) and

44. Properties of the Exponential Function f(x) = ax, a > 1

45. Properties of the Exponential Function • Properties of f(x) = ax, 0 <a < 1 • Domain: All real numbers • Range: Positive real numbers; (0, 1) • Intercepts: • No x-intercepts • y-intercept of y = 1 • x-axis is horizontal asymptote as x 1 • Decreasing and one-to-one. • Smooth and continuous • Contains points (0,1), (1, a) and

46. Properties of the Exponential Function f(x) = ax, 0 <a < 1

47. The Number e • Number e: the number that the expression approaches as n1. • Use ex or exp(x) on your calculator.

48. The Number e • Estimating value of e • n = 1: 2 • n = 2: 2.25 • n = 5: 2.488 32 • n = 10: 2.593 742 460 1 • n = 100: 2.704 813 829 42 • n = 1000: 2.716 923 932 24 • n = 1,000,000,000: 2.718 281 827 10 • n = 1,000,000,000,000: 2.718 281 828 46

49. Exponential Equations • If au = av, then u = v • Another way of saying that the function f(x) = ax is one-to-one. • Examples. (a) Problem: Solve 23x {1 = 32 Answer: (b) Problem: Solve Answer:

50. Key Points • Exponents • Laws of Exponents • Exponential Functions • Graphing Exponential Functions • Properties of the Exponential Function • The Number e • Exponential Equations