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EXAMPLE 1

5. 3. x +. = y. 1. 3. EXAMPLE 1. Find an inverse relation. Find an equation for the inverse of the relation y = 3 x – 5. y = 3 x – 5. Write original relation. x = 3 y – 5. Switch x and y. x + 5 = 3 y. Add 5 to each side. Solve for y . This is the inverse relation. x.

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EXAMPLE 1

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  1. 5 3 x+ =y 1 3 EXAMPLE 1 Find an inverse relation Find an equation for the inverse of the relation y = 3x – 5. y = 3x – 5 Write original relation. x = 3y – 5 Switch x and y. x + 5 = 3y Add 5 to each side. Solve for y. This is the inverse relation.

  2. x + Verify thatf(x) = 3x – 5 and f –1(x) = 5 5 1 1 5 5 are inverse functions. 3 3 3 3 3 3 = x 1 1 5 5 x + x + f (f –1(x)) =f f –1(f(x)) = f –1(3x – 5) 3 3 3 3 (3x – 5) + = – 5 = 3 = x – + = x EXAMPLE 2 Verify that functions are inverses SOLUTION STEP 1 STEP 2 Show: that f(f –1(x)) = x. Show: that f –1(f(x)) = x. = x + 5 – 5

  3. Fitness 3 8 Elastic bands can be used in exercising to provide a range of resistance. A band’s resistance R (in pounds) can be modeled by R = L – 5 where Lis the total length of the stretched band (in inches). EXAMPLE 3 Solve a multi-step problem

  4. 8 3 3 • Find the inverse of the model. 3 8 8 • Use the inverse function to find the length at which the band provides 19pounds of resistance. L – 5 R = L R + 5 = 40 8 . R + = L Multiply each side by 3 3 EXAMPLE 3 Solve a multi-step problem SOLUTION STEP 1 Find: the inverse function. Write original model. Add 5 to each side.

  5. 40 192 152 8 40 8 40 = (19) + + R + = L = = + 3 3 3 3 3 3 3 ANSWER The band provides 19pounds of resistance when it is stretched to 64 inches. EXAMPLE 3 Solve a multi-step problem STEP 2 Evaluate: the inverse function when R = 19. = 64

  6. for Examples 1, 2, and 3 GUIDED PRACTICE Find the inverse of the given function. Then verify that your result and the original function are inverses. 1. f(x) = x + 4 y = x + 4 Write original relation. x = y + 4 Switch x and y. x – 4 = y Subtract 4 from each side.

  7. x + 1 =y 2 for Examples 1, 2, and 3 GUIDED PRACTICE 2. f(x) = 2x – 1 y = 2x – 1 Write original relation. x = 2y – 1 Switch x and y. x + 1 = 2y Add 1 to each side. Divide both sides by 2.

  8. x 1 =y 3 for Examples 1, 2, and 3 GUIDED PRACTICE 3. f(x) = –3x – 1 y = –3x + 1 Write original relation. x = –3y +1 Switch x and y. x – 1 = –3y Subtract 1 to each side. Solve for y. This is the inverse relation.

  9. 40 3 ANSWER 8 40 8 The band provides 13pounds of resistance when it is stretched to 48 inches. (13) + R + L = = + 3 3 3 for Examples 1, 2, and 3 GUIDED PRACTICE 4. Fitness: Use the inverse function in Example 3 to find the length at which the band provides 13pounds of resistance. SOLUTION Evaluate the inverse function when R = 3 = 48

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