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5–Minute Check 1

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

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5–Minute Check 1

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  1. A. B. C. D. Given f(x) = 3x and g(x) = x2 – 1, find (f ● g)(x)and its domain. 5–Minute Check 1

  2. A. B. C. D. Given f(x) = 3x and g(x) = x2 – 1, find (f ● g)(x)and its domain. 5–Minute Check 1

  3. Given f(x) = 3x and g(x) = x2 – 1, find and its domain. A. B. C. D. 5–Minute Check 2

  4. Given f(x) = 3x and g(x) = x2 – 1, find and its domain. A. B. C. D. 5–Minute Check 2

  5. A. B. C. D. Given f(x) = 3x and g(x) = x2 – 1, find [g ○ f](x) and its domain. 5–Minute Check 3

  6. A. B. C. D. Given f(x) = 3x and g(x) = x2 – 1, find [g ○ f](x) and its domain. 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)= 4x2 + 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)= 4x2+ 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) = x5+ x3– 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)= x5 + x3– 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 graphingcalculator, and apply the horizontal line test to determine whether its inverse function exists. Write yes or no. C. no D. no A. yes B. yes Example 1

  12. Key Concept 2

  13. 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.   Find Inverse Functions Algebraically Example 2

  14. Find Inverse Functions Algebraically Example 2

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

  18. 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; Find Inverse Functions Algebraically Example 2

  19. 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. Find Inverse Functions Algebraically Example 2

  20. 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–1has 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, . Find Inverse Functions Algebraically Add 1 to each side. Replace y with f–1(x). Example 2

  22. Find Inverse Functions Algebraically Answer: Example 2

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

  24. 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 . 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. 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) = x2 – 2, x 0 and are inverses of each other. A. B. C. D. Example 3

  32. Show that f(x) = x2 – 2, x 0 and are inverses of each other. A. B. C. D. Example 3

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

  37. A. B. C. D. Use the graph of the function to graph its inverse function. 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) = 96,000 + 80x. Explain why the inverse function f –1(x) exists. Then find f –1(x). The graph of f(x) = 96,000 + 80x passes the horizontal line test. Therefore, f(x) is a one-to one function and has an inverse function. Example 5

  39. Use an Inverse Function f(x) = 96,000 + 80x Original 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). 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 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) = 96,000 + 80x. What do f–1(x) and x represent in the inversefunction? 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) = 96,000 + 80x. What do f–1(x) and x represent in the inversefunction? 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) = 96,000 + 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. 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. Because , the number of stereos made for a total cost of $216,000 is 1500. 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. Answer: Example 5

  48. Because , the number of stereos made for a total cost of $216,000 is 1500. 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. 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) = 480 + 0.05x. Explain why the inverse function f –1(x) exists. Then find f –1(x). Example 5

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

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