Mathematics 116 Chapter 5

1. Mathematics 116Chapter 5 • Exponential • And • Logarithmic Functions

2. John Quincy Adams • “Patience and perseverance have a magical effect before which difficulties disappear and obstacles vanish.” • Mathematics 116 • Exponential Functions • and • Their Graphs

3. Def: Relation • A relation is a set of ordered pairs. • Designated by: • Listing • Graphs • Tables • Algebraic equation • Picture • Sentence

4. Def: Function • A function is a set of ordered pairs in which no two different ordered pairs have the same first component. • Vertical line test – used to determine whether a graph represents a function.

5. Defs: domain and range • Domain: The set of first components of a relation. • Range: The set of second components of a relation

6. Examples of Relations:

7. Objectives • Determine the domain, range of relations. • Determine if relation is a function.

8. Mathematics 116 • Inverse Functions

9. Objectives: • Determine the inverse of a function whose ordered pairs are listed. • Determine if a function is one to one.

10. Inverse Function • g is the inverse of f if the domains and ranges are interchanged. • f = {(1,2),(3,4), (5,6)} • g= {(2,1), (4,3),(6,5)}

11. Inverse of a function

12. Inverse of function

13. One-to-One Function • A function f is one-to one if for and and b in its domain, f(a) = f(b) implies a = b. • Other – each component of the range is unique.

14. One-to-One function • Def: A function is a one-to-one function if no two different ordered pairs have the same second coordinate.

15. Horizontal Line TestA test for one-to one • If a horizontal line intersects the graph of the function in more than one point, the function is not one-to one

16. Existence of an Inverse Function • A function f has an inverse function if and only if f is one to one.

17. Find an Inverse Function • 1. Determine if f has an inverse function using horizontal line test. • 2. Replace f(x) with y • 3. Interchange x and y • 4. Solve for y • 5. Replace y with

18. Definition of Inverse Function • Let f and g be two functions such that f(g(x))=x for every x in the domain of g and g(f(x))=x for every x in the domain of. • g is the inverse function of the function f

19. Objective • Recognize and evaluate exponential functions with base b.

20. Michael Crichton – The Andromeda Strain (1971) • The mathematics of uncontrolled growth are frightening. A single cell of the bacterium E. coli would, under ideal circumstances, divide every twenty minutes. It this way it can be shown that in a single day, one cell of E. coli could produce a super-colony equal in size and weight to the entire planet Earth.”

21. Graph • Determine: Domain, range, function and why, 1-1 function and why, y intercept, x intercept, asymptotes

22. Graph • Determine: Domain, range, function and why, 1-1 function and why, y intercept, x intercept, asymptotes

23. Exponential functions • Exponential growth • Exponential decay

24. Properties of graphs of exponential functions • Function and 1 to 1 • y intercept is (0,1) and no x intercept(s) • Domain is all real numbers • Range is {y|y>0} • Graph approaches but does not touch x axis – x axis is asymptote • Growth or decay determined by base

25. The Natural Base e

26. The natural base e

27. Calculator Keys • Second function of divide • Second function of LN (left side)

28. Dwight Eisenhower – American President • “Pessimism never won any battle.”

29. Property of equivalent exponents • For b>0 and b not equal to 1

30. Compound Interest • A = Amount • P = Principal • r = annual interest rate in decimal form • t= number of years

31. Continuous Compounding • A = Amount • P = Principal • r = rate in decimal form • t = number of years

32. Compound interest problem • Find the accumulated amount in an account if $5,000 is deposited at 6% compounded quarterly for 10 years.

33. Objectives • Recognize and evaluate exponential functions with base b • Graph exponential functions • Recognize, evaluate, and graph exponential functions with base e. • Use exponential functions to model and solve real-life problems.

34. Albert Einstein – early 20th century physicist • “Everything should be made as simple as possible, but not simpler.”

35. Mathematics 116 – 4.2 • Logarithmic Functions • and • Their Graphs

36. Definition of Logarithm

37. Objectives • Recognize and evaluate logarithmic function with base b • Note: this includes base 10 and base e • Graph logarithmic functions • By Hand • By Calculator

38. Shape of logarithmic graphs • For b > 1, the graph rises from left to right. • For 0 < b < 1, the graphs falls from left to right.

39. Properties of Logarithmic Function • Domain:{x|x>0} • Range: all real numbers • x intercept: (1,0) • No y intercept • Approaches y axis as vertical asymptote • Base determines shape.

40. Evaluate Logs on calculator • Common Logs – base of 10 • Natural logs – base of e

41. Basic Properties of logs

42. **Property of Logarithms • One to One Property

43. Objective • Use logarithmic functions to model and solve real-life problems.

44. Jim Rohn • “You must take personal responsibility. You cannot change the circumstances, the seasons, or the wind, but you can change yourself. That is something you have charge of.”

45. Mathematics 116 – 4.3 • Properties • of • Logarithms

46. Change of Base Formula

47. Problem: change of base

48. Logarithm Theorems

49. Basic Properties of logarithms

50. For x>0, y>0, b>0 and b not 1Product rule of Logarithms