Transcription of NAME DATE PERIOD 6-2 Study Guide and Intervention
1 NAME DATE PERIOD . 6- 2 study guide and intervention Inverse Functions and Relations Find Inverses Two relations are inverse relations if and only if whenever one relation contains the Inverse Relations element (a, b), the other relation contains the element (b, a). Property of Inverse Suppose f and f -1 are inverse functions. Functions Then f(a) = b if and only if f -1(b) = a. Example 2. Find the inverse of the function f(x) = 1. x- . Then graph the 5 5. function and its inverse. Step 1 Replace f (x) with y in the original equation. f (x ). 4. 2 1 2 1. f(x) = x- y= x- 2. 5 5 5 5. f (x) = 2 5x - 1 5.
2 Step 2 Interchange x and y. -4 -2 O 2 4x 2 1. x= y- -2. 5 5. f 1(x) = 5 2x + 1 2. Step 3 Solve for y. -4. 2 1 2 1. x= y- Inverse of y = x- . 5 5 5 5. 5x = 2y - 1 Multiply each side by 5. 5x + 1 = 2y Add 1 to each side. 1. (5x + 1) = y Divide each side by 2. 2. Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 2 1 1. The inverse of f(x) = x- is f -1(x) = (5x + 1). 5 5 2. Exercises Find the inverse of each function. Then graph the function and its inverse. 2 1. 1. f(x) = x-1 2. f(x) = 2x - 3 3. f(x) = x-2. 3 4. 3 3 1 3. f -1(x) = x+ f -1(x) = x+ f -1(x) = 4x + 8.
3 2 2 2 2. f (x) f(x) f(x). 4 4 4. f 1(x) = 3 2x + 3 2 f -1(x) = 1 2x + 3 2 1. 2 f (x) = 4x - 2. -4 -2 O 2 4 x -4 -2 O 2 4x -4 -2 O 2 4x -2. -2 f(x) = 2 x - 1 -2. 3. -4. -4 -4 f(x) = 2x - 3 f -1(x) = 4x + 8. Chapter 6 12 Glencoe Algebra 2. NAME DATE PERIOD . 6- 2 study guide and intervention (continued). Inverse Functions and Relations Verifying Inverses Inverse Functions Two functions f(x) and g(x) are inverse functions if and only if [f g](x) = x and [g f ](x) = x. Example 1 1. Determine whether f(x) = 2x - 7 and g(x) = (x + 7) are 2. inverse functions. [ f g](x) = f[ g(x)] [ g f ](x) = g[ f(x)].
4 1. = f (x + 7) = g(2x - 7). 2. 1 1. 2 . = 2 (x + 7) - 7 = (2x - 7 + 7). 2. =x+7-7 =x =x The functions are inverses since both [ f g](x) = x and [ g f ](x) = x. Example 2 1. Determine whether f(x) = 4x + 1. and g(x) = x - 3 are Lesson 6-2. 3 4. inverse functions. [ f g](x) = f[ g(x)]. (4 ). 1. =f x-3. = 4( . 1x - 3. 4 ) + 13. Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 1. = x - 12 + . 3. 2. = x - 11 . 3. Since [ f g](x) x, the functions are not inverses. Exercises Determine whether each pair of functions are inverse functions. Write yes or no. 1 1.
5 1. f(x) = 3x - 1 2. f(x) = x+5 3. f(x) = x - 10. 4 2. 1 1 1 no g(x) = x+ yes g(x) = 4x - 20 yes g(x) = 2x + . 3 3 10. 4. f(x) = 2x + 5 5. f(x) = 8x - 12 6. f(x) = -2x + 3. 1 1 3 yes g(x) = 5x + 2 no g(x) = x + 12 no g(x) = - x+ . 8 2 2. 1 3 1. 7. f(x) = 4x - 8. f(x) = 2x - 9. f(x) = 4x + . 2 5 2. 1 1 1 1 3. g(x) = x + yes g(x) = (5x + 3) yes g(x) = x - no 4 8 10 2 2. x 4 3. 10. f(x) = 10 - 11. f(x) = 4x - 12. f(x) = 9 + x 2 5 2. x 1 2 - g(x) = 20 - 2x yes g(x) = + . 4 5. yes g(x) = x 6. 3. yes Chapter 6 13 Glencoe Algebra 2.
