1) An inverse function f^-1 interchanges the x and y values of the original function f. 2) For a function to have an inverse, it must be one-to-one, meaning each x maps to a single y and vice versa. 3) To find the inverse of a function algebraically, interchange x and y and solve for y. The graph of an inverse is the reflection of the original graph across the line y=x.

1. Inverse Functions and Relations

2. Inverse Functions and Relations

4. Let f be defined as the set of values given by Let f -1 be defined as the set of values given by 10 -5 4 0 y-values 7 4 0 -2 x-values 7 4 0 -2 y-values 10 -5 4 0 x-values

9. Example: Inverse Relation Algebraically Example1 : Find the inverse relation algebraically for the function f ( x ) = 3 x + 2. y = 3 x + 2 Original equation defining f x = 3 y + 2 Switch x and y . 3 y + 2 = x Reverse sides of the equation. To calculate a value for the inverse of f , subtract 2, then divide by 3 . To find the inverse of a relation algebraically , interchange x and y and solve for y . y -1 = Solve for y.

10. y = x The graphs of a relation and its inverse are reflections in the line y = x . The ordered pairs of f a re given by the equation . Example 1a : Find the graph of the inverse relation geometrically from the graph of f ( x ) = x y 2 -2 -2 2 The ordered pairs of the inverse are given by .

12. Example 3: Find the inverse of f(x) = f -1 (x) = -2x +2 (-2) (-2) Replace f(x) with y. Interchange x and y. Solve for y. Replace y with f-1(x).

13. Example 4: Two functions f and g are inverse functions if and only if both of their compositions are the identity function; f(x) = x. Determine whether and are inverse functions. You must do [f ◦ g](x) and [g ◦ f ](x), if they both equal x, they are inverses!

14. [f ◦ g](x) = x + 6 – 6 = x [g ◦ f ](x) = x – 8 + 8 = x So, they ARE inverses of each other!

15. Example: Composition of Functions It follows that g = f -1 . Example 5 :Verify that the function g ( x ) = is the inverse of f ( x ) = 2 x – 1. f( g ( x ) ) = 2 g ( x ) – 1 = 2( ) – 1 = ( x + 1) – 1 = x g ( f ( x ) ) = = = = x