Inverse Relations and Functions LESSON 1–7 Inverse Relations and Functions

Key Concept: Horizontal Line Test Five-Minute Check TEKS Then/Now New Vocabulary Key Concept: Horizontal Line Test Example 1: Apply the Horizontal Line Test Key Concept: Finding an Inverse Function Example 2: Find Inverse Functions Algebraically Key Concept: Compositions of Inverse Functions Example 3: Verify Inverse Functions Example 4: Find Inverse Functions Graphically Example 5: Real-World Example: Use an Inverse Function Lesson Menu

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

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

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

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

Targeted TEKS P.2(E) Determine an inverse function, when it exists, for a given function over its domain or a subset of its domain and represent the inverse using multiple representations. Mathematical Processes P.1(B), P.1(C)

You found the composition of two functions. (Lesson 1-6) Use the graphs of functions to determine if they have inverse functions. Find inverse functions algebraically and graphically. Then/Now

inverse relation inverse function one-to-one Vocabulary

Key Concept 1

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 Example 1

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 Example 1

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

Key Concept 2

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

Find Inverse Functions Algebraically Example 2

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

Find Inverse Functions Algebraically Divide. Example 2

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

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

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

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

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

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

Key Concept 3

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

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

Verify Inverse Functions Answer: Example 3

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

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

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

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

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

f (x) = 96,000 + 80x Original function Use an Inverse Function f (x) = 96,000 + 80x Original function y = 96,000 + 80x Replace f (x) with y. x = 96,000 + 80y 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

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

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. Example 5

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

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

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 Example 5

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

A. B. C. D. Example 5

Inverse Relations and Functions LESSON 1–7 Inverse Relations and Functions