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

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

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

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

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

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

7. inverse relation
inverse function
one-to-one
Vocabulary

8. Key Concept 1

9. 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:
Example 1

10. 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:
Example 1

11. 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
Example 1

12. Key Concept 2

13. A. Determine whether f has an inverse function for
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. 
Example 2

14. Find Inverse Functions Algebraically
Example 2

15. 2xy – y = x Isolate the y-terms. y(2x –1) = x Factor.
Find Inverse Functions Algebraically
Original function
Replace f(x) with y.
Interchange x and y.
2xy – x = y Multiply each side by 2y – 1. Then apply the Distributive Property.
2xy – y = x Isolate the y-terms.
y(2x –1) = x Factor.
Example 2

16. Find Inverse Functions Algebraically
Divide.
Example 2

17. Find Inverse Functions Algebraically
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. 
Answer:
Example 2

18. Find Inverse Functions Algebraically
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. 
Answer: f –1 exists;
Example 2

19. 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.
Example 2

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

21. From the graph, you can see that f –1 has domain
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,
Example 2

22. Find Inverse Functions Algebraically
Answer:
Example 2

23. Answer: f –1 exists with domain ;
Find Inverse Functions Algebraically
Answer: f –1 exists with domain ;
Example 2

24. Determine whether f has an inverse function for
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.
Example 2

25. Determine whether f has an inverse function for
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.
Example 2

26. Key Concept 3

27. Show that f [g (x)] = x and g [f (x)] = x.
Verify Inverse Functions
Show that f [g (x)] = x and g [f (x)] = x.
Example 3

28. 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.
Example 3

29. Verify Inverse Functions
Answer:
Example 3

30. Verify Inverse Functions
Answer:
Example 3

31. Show that f (x) = x 2 – 2, x  0 and are inverses of each other.B. C. D.
Example 3

32. Show that f (x) = x 2 – 2, x  0 and are inverses of each other.B. C. D.
Example 3

33. Use the graph of relation A to sketch the graph of its inverse.
Find Inverse Functions Graphically
Use the graph of relation A to sketch the graph of its inverse.
Example 4

34. Find Inverse Functions Graphically
Graph the line y = x. Locate a few points on the graph of f (x). Reflect these points in y = x. Then connect them with a smooth curve that mirrors the curvature of f (x) in line y = x.
Answer:
Example 4

35. Find Inverse Functions Graphically
Graph the line y = x. Locate a few points on the graph of f (x). Reflect these points in y = x. Then connect them with a smooth curve that mirrors the curvature of f (x) in line y = x.
Answer:
Example 4

36. Use the graph of the function to graph its inverse function.B. C. D.
Example 4

37. Use the graph of the function to graph its inverse function.B. C. D.
Example 4

38. Use an Inverse Function
A. 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, x. Explain why the inverse function f –1(x) exists. Then find f –1(x).
The graph of f (x) = 96, x passes the horizontal line test. Therefore, f (x) is a one-to one function and has an inverse function.
Example 5

39. f (x) = 96,000 + 80x Original function
Use an Inverse Function
f (x) = 96, x Original function
y = 96, x Replace f (x) with y.
x = 96, y Interchange x and y.
x – 96,000 = 80y Subtract 96,000 from each side.
Divide each side by 80.
Replace y with f –1(x).
Example 5

40. Use an Inverse Function
Answer:
Example 5

41. Answer: The graph of f (x) passes the horizontal line test.
Use an Inverse Function
Answer: The graph of f (x) passes the horizontal line test.
Example 5

42. 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, x. 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:
Example 5

43. 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, x. 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.
Example 5

44. 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, x. 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.
Example 5

45. Use an Inverse Function
Answer:
Example 5

46. Use an Inverse Function
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.
Example 5

47. 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, x. 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:
Example 5

48. 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, x. 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
Example 5

49. 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) = x. Explain why the inverse function f –1(x) exists. Then find f –1(x).
Example 5

50. A. B. C. D.
Example 5

51. A. B. C. D.
Example 5