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Lesson Menu Five-Minute Check 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

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.

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

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.

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

Then/Now 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.

Vocabulary inverse relation inverse function one-to-one

Key Concept 1

Example 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

Key Concept 2

Example 2 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 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.

Example 2 Find Inverse Functions Algebraically Divide.

Example 2 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. 

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

Example 2 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.

Key Concept 3

Example 3 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. A. B. C.D.

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

Example 4 Answer: 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.

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

Example 5 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 Use an Inverse Function f (x)= 96, xOriginal function y= 96, xReplace f (x) with y. x= 96, yInterchange 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 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, 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 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 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

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

Example 5 A. B. C. D.

End of the Lesson