( ). Switch x and y and solve for y:
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1 . Let y = f ( x). Switch x and y and solve for y: y! = x y = x + y = x + The inverse is f! (x) = x +.. Let y = f x y = x y = The inverse is f! (x) = 5. Let y = f x 8.2 The Inverse of a Function x x. y! y! = x y! = xy! x y! xy =! x ( ) =! x y! x y =! x! x = x! x! The inverse is f! (x) = x! x!. 7. Let y = f x y! = x 4 y! = 4x y = 4x + The inverse is f! (x) = 4x +.
2 9. Let y = f ( x). Switch x and y and solve for y: 2 y! = x y! 6 = 2x y = 2x + 6 The inverse is f! (x) = 2x Let y = f ( x). Switch x and y and solve for y: 2 y! = x 2y! 9 = x 2y = x + 9 y = 2 x The inverse is f! (x) = 2 x Let y = f ( x). Switch x and y and solve for y: y! 4 = x y = x + 4 y = x + 4 The inverse is f! (x) = x Let y = f x 4y! 2y + = x 4y! = 2xy + x 4y! 2xy = x + ( ) = x + y 4! 2x y = x + 4! 2x The inverse is f! (x) = x + 4! 2x..
3 7. Let y = f ( x). Switch x and y and solve for y: 2y + y + = x 2y + = xy + x 2y! xy = x! ( ) = x! y 2! x y = x! 2! x =! x x! 2 The inverse is f! (x) =! x x! Finding the inverse: 2. Finding the inverse: 2y! = x y 2! = x 2y = x + y = x + 2 y 2 = x + y = ± x + The inverse is y! = x + 2. Graphing: The inverse is y! = ± x +. Graphing:
4 2. Finding the inverse: y 2! 2y! = x y 2! 2y + = x + + ( y! ) 2 = x + 4 y! = ± x + 4 y = ± x + 4 The inverse is y! = ± x + 4. Graphing: 25. The inverse is x = y. Graphing each curve: 27. The inverse is x = 4. Graphing each curve:
5 29. Finding the inverse:. Finding the inverse: 2 y = x 2 y + 2 = x y = 2x y + 4 = 2x y = 2x The inverse is y! = y = 2x! 4 2x. Graphing: The inverse is y! = 2x! 4. Graphing:. Finding the inverse: y + 2 = x y + 2 = x 2 y = x 2! 2 The inverse is y! = x 2! 2, x " 0. Graphing each curve: 5. a. Yes, this function is one-to-one. b. No, this function is not one-to-one. c. Yes, this function is one-to-one.
6 7. a. Evaluating the function: f ( 2) = ( 2)! 2 = 6! 2 = 4 b. Evaluating the function: f! ( 2) = = 4 c. Evaluating the function: f "# f! ( 2) $ % = f d. Evaluating the function: f! "# f ( 2) $% = f! ( 4) = Let y = f ( x). Switch x and y and solve for y: y = x & ' ( 4 ) * + = & 4 ) ' ( * +! 2 = 4! 2 = 2 = 6 = 2 y = x The inverse is f! (x) = x. 4. The inverse is f! (x) = 7(x + 2). 4. a. The value is. b. The value is 6. c. The value is 2. d. The value is. e. The value is 2. f. The value is. g. Each is an inverse of the other. 45. a. Substituting t = 5: s( 5) = 6( 5) = $489.4 billion b. Finding the inverse: s = 6t t = s! t(s) = s! ! c. Substitute s = 507: t(507) = " 6. 6 The payments will reach $507 billion in the year 2006.
7 47. a. Substituting m = 4520: f = b. Finding the inverse: f = 22m 5 5 f = 22m m( f ) = 5 f 22 c. Substituting f = 2: m(2) = Simplifying:!2 = = Solving the equation: 2 = x 22 ( 4520 )! 6629 feet per second 5 ( ) x = 2 5. Solving the equation: 4 = x x = Completing the statement: 8 = Completing the statement: 0,000 = Completing the statement: 8 = 4 6. Completing the statement: 6 = 6!.6 mph
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