1. One-to-One Functions Inverse Function

2. A function f is one-to-one if for each x in the domain of f there is exactly one y in the range and no y in the range is the image of more than one x in the domain. A function is not one-to-one if two different elements in the domain correspond to the same element in the range.

3. x1x1 x2x2 x3x3 y1y1 y2y2 y3y3 x1x1 x3x3 y1y1 y2y2 y3y3 x1x1 x2x2 x3x3 y1y1 y3y3 Domain Range One-to-one function Not a function NOT One-to-one function

4. M : Mother Function is NOT one-one Joe Samantha Anna Ian Chelsea George Laura Julie Hilary Barbara Sue HumansMothers

5. S: Social Security function IS one-one Joe Samantha Anna Ian Chelsea George 123456789 223456789 333456789 433456789 533456789 633456789 AmericansSSN

6. Is the function f below one – one? 12345671234567 10 11 12 13 14 15 16

7. Theorem Horizontal Line Test If horizontal lines intersect the graph of a function f in at most one point, then f is one-to-one.

8. Use the graph to determine whether the function is one-to-one. Not one-to-one.

9. Use the graph to determine whether the function is one-to-one. One-to-one.

10. The inverse of a one-one function is obtained by switching the role of x and y a one-to-one function that "reverses" another one-to-one function, defines an inverse function.

11. 123456789 223456789 333456789 433456789 533456789 633456789 SSN Joe Samantha Anna Ian Chelsea George Americans The inverse of the social security function

13. Let f denote a one-to-one function y = f(x). The inverse of f, denoted by f -1, is a function such that f -1 (f( x )) = x for every x in the domain of f and f(f -1 (x))=x for every x in the domain of f -1.

14. Domain of fRange of f

15. Theorem The graph of a function f and the graph of its inverse are symmetric with respect to the line y = x.

16. y = x (2, 0) (0, 2)

17. Finding the inverse of a 1-1 function Step1: Write the equation in the form Step2: Interchange x and y. Step 3: Solve for y. Step 4: Write for y.

18. Find the inverse of Step1: Step2: Interchange x and y Step 3: Solve for y

22. 5–Minute Check 1 Given f (x) = 3x and g (x) = x 2 – 1, find (f ● g)(x) and its domain. A. B. C. D.

23. 5–Minute Check 1 Given f (x) = 3x and g (x) = x 2 – 1, find (f ● g)(x) and its domain. A. B. C. D.

24. 5–Minute Check 2 Given f (x) = 3x and g (x) = x 2 – 1, find and its domain. A. B. C. D.

25. 5–Minute Check 2 Given f (x) = 3x and g (x) = x 2 – 1, find and its domain. A. B. C. D.

26. 5–Minute Check 3 Given f (x) = 3x and g (x) = x 2 – 1, find [g ○ f](x) and its domain. A. B. C. D.

27. 5–Minute Check 3 Given f (x) = 3x and g (x) = x 2 – 1, find [g ○ f](x) and its domain. A. B. C. D.

28. 5–Minute Check 4 Find two functions f and g such that h (x) = [f ○ g](x). A. B. C. D.

29. 5–Minute Check 4 Find two functions f and g such that h (x) = [f ○ g](x). A. B. C. D.

30. You found the composition of two functions. (Semester 1) Use the graphs of functions to determine if they have inverse functions. Find inverse functions algebraically and graphically.

31. Key Concept 1

32. Apply the Horizontal Line Test A. Graph the function f (x) = 4x 2 + 4x + 1 using a graphing calculator, and apply the horizontal line test to determine whether its inverse function exists. Write yes or no. The graph of f (x) = 4x 2 + 4x + 1 shows that it is possible to find a horizontal line that intersects the graph of f (x) more than once. Therefore, you can conclude that f –1 does not exist. Answer:

33. Apply the Horizontal Line Test A. Graph the function f (x) = 4x 2 + 4x + 1 using a graphing calculator, and apply the horizontal line test to determine whether its inverse function exists. Write yes or no. The graph of f (x) = 4x 2 + 4x + 1 shows that it is possible to find a horizontal line that intersects the graph of f (x) more than once. Therefore, you can conclude that f –1 does not exist. Answer: no

34. Apply the Horizontal Line Test B. Graph the function f (x) = x 5 + x 3 – 1 using a graphing calculator, and apply the horizontal line test to determine whether its inverse function exists. Write yes or no. The graph of f (x) = x 5 + x 3 – 1 shows that it is not possible to find a horizontal line that intersects the graph of f (x) more than one point. Therefore, you can conclude that f –1 exists. Answer:

35. Apply the Horizontal Line Test B. Graph the function f (x) = x 5 + x 3 – 1 using a graphing calculator, and apply the horizontal line test to determine whether its inverse function exists. Write yes or no. The graph of f (x) = x 5 + x 3 – 1 shows that it is not possible to find a horizontal line that intersects the graph of f (x) more than one point. Therefore, you can conclude that f –1 exists. Answer: yes

36. Graph the function using a graphing calculator, and apply the horizontal line test to determine whether its inverse function exists. Write yes or no. A.yes B.yes C.no D.no

37. Graph the function using a graphing calculator, and apply the horizontal line test to determine whether its inverse function exists. Write yes or no. A.yes B.yes C.no D.no

38. Key Concept 2

39. Find Inverse Functions Algebraically A. Determine whether f has an inverse function for. If it does, find the inverse function and state any restrictions on its domain. The graph of f passes the horizontal line test. Therefore, f is a one-one function and has an inverse function. From the graph, you can see that f has domain and range. Now find f – 1. UU

40. Find Inverse Functions Algebraically Original function Replace f(x) with y. Interchange x and y. 2xy – x= yMultiply each side by 2y – 1. Then apply the Distributive Property. 2xy – y= xIsolate the y-terms. y(2x –1) = xFactor.

41. Find Inverse Functions Algebraically Divide.

42. Find Inverse Functions Algebraically Answer: From the graph, you can see that f – 1 has domain and range. The domain and range of f is equal to the range and domain of f – 1, respectively. Therefore, it is not necessary to restrict the domain of f – 1. UU

43. Find Inverse Functions Algebraically Answer: f –1 exists; From the graph, you can see that f – 1 has domain and range. The domain and range of f is equal to the range and domain of f – 1, respectively. Therefore, it is not necessary to restrict the domain of f – 1. UU

44. Find Inverse Functions Algebraically B. Determine whether f has an inverse function for. If it does, find the inverse function and state any restrictions on its domain. The graph of f passes the horizontal line test. Therefore, f is a one-one function and has an inverse function. From the graph, you can see that f has domain and range. Now find f – 1.

45. Find Inverse Functions Algebraically Original function Replace f(x) with y. Interchange x and y. Divide each side by 2. Square each side.

46. Find Inverse Functions Algebraically Add 1 to each side. Replace y with f – 1 (x). From the graph, you can see that f – 1 has domain and range. By restricting the domain of f – 1 to, the range remains. Only then are the domain and range of f equal to the range and domain of f –1, respectively. So,.

47. Find Inverse Functions Algebraically Answer:

48. Find Inverse Functions Algebraically Answer: f –1 exists with domain ;

49. Determine whether f has an inverse function for. If it does, find the inverse function and state any restrictions on its domain. A. B. C. D.f –1 (x) does not exist.

50. Determine whether f has an inverse function for. If it does, find the inverse function and state any restrictions on its domain. A. B. C. D.f –1 (x) does not exist.

51. Key Concept 3

52. Verify Inverse Functions Show that f [g (x)] = x and g [f (x)] = x.

53. Verify Inverse Functions Because f [g (x)] = x and g [f (x)] = x, f (x) and g (x) are inverse functions. This is supported graphically because f (x) and g (x) appear to be reflections of each other in the line y = x.

54. Verify Inverse Functions Answer:

55. Verify Inverse Functions Answer:

56. Use an Inverse Function f (x)= 96,000 + 80xOriginal function y= 96,000 + 80xReplace f (x) with y. x= 96,000 + 80yInterchange x and y. x – 96,000= 80ySubtract 96,000 from each side. Divide each side by 80. Replace y with f –1 (x).

57. Use an Inverse Function Answer:

58. Use an Inverse Function Answer:The graph of f (x) passes the horizontal line test.

59. Use an Inverse Function B. MANUFACTURING The fixed costs for manufacturing one type of stereo system are $96,000 with variable cost of $80 per unit. The total cost f (x) of making x stereos is given by f (x) = 96,000 + 80x. What do f –1 (x) and x represent in the inverse function? In the inverse function, x represents the total cost and f –1 (x) represents the number of stereos. Answer:

60. Use an Inverse Function B. MANUFACTURING The fixed costs for manufacturing one type of stereo system are $96,000 with variable cost of $80 per unit. The total cost f (x) of making x stereos is given by f (x) = 96,000 + 80x. What do f –1 (x) and x represent in the inverse function? In the inverse function, x represents the total cost and f –1 (x) represents the number of stereos. Answer:In the inverse function, x represents the total cost and f –1 (x) represents the number of stereos.

61. Use an Inverse Function C. MANUFACTURING The fixed costs for manufacturing one type of stereo system are $96,000 with variable cost of $80 per unit. The total cost f (x) of making x stereos is given by f (x) = 96,000 + 80x. What restrictions, if any, should be placed on the domain of f (x) and f –1 (x)? Explain. The function f (x) assumes that the fixed costs are nonnegative and that the number of stereos is an integer. Therefore, the domain of f(x) has to be nonnegative integers. Because the range of f (x) must equal the domain of f –1 (x), the domain of f –1 (x) must be multiples of 80 greater than 96,000.

62. Answer: Use an Inverse Function

63. Answer:The domain of f (x) has to be nonnegative integers. The domain of f –1 (x) is multiples of 80 greater than 96,000. Use an Inverse Function

64. D. MANUFACTURING The fixed costs for manufacturing one type of stereo system are $96,000 with variable cost of $80 per unit. The total cost f (x) of making x stereos is given by f (x) = 96,000 + 80x. Find the number of stereos made if the total cost was $216,000. Because, the number of stereos made for a total cost of $216,000 is 1500. Answer:

65. Use an Inverse Function D. MANUFACTURING The fixed costs for manufacturing one type of stereo system are $96,000 with variable cost of $80 per unit. The total cost f (x) of making x stereos is given by f (x) = 96,000 + 80x. Find the number of stereos made if the total cost was $216,000. Because, the number of stereos made for a total cost of $216,000 is 1500. Answer:1500 stereos

66. EARNINGS Ernesto earns $12 an hour and a commission of 5% of his total sales as a salesperson. His total earnings f (x) for a week in which he worked 40 hours and had a total sales of $x is given by f (x) = 480 + 0.05x. Explain why the inverse function f –1 (x) exists. Then find f –1 (x).

67. A. B. C. D.

68. A. B. C. D.