1.6 Inverse Functions

Inverse: found by interchanging the x & y coordinates * the inverse of a relation is a relation * but the inverse of a function may or may not be a function Ex 1) Is the relation a function? Find inverse. Is inverse a function? {(2, a), (2, b), (3, d), (3, e)} - Not a function Inverse: - Yes, a function two 2 inputs, two 3 inputs {(a, 2), (b, 2), (d, 3), (e, 3)}

Finding inverse of a function: Def: g is the inverse function of f if and only if f (g(x)) = x for all x in domain of g and g(f (x)) = x for all x in domain of f Notation: Finding inverse of a function: write as y = switch x & y solve for y The graph will be the reflection over the line y = x

Ex 2) Find inverse. Graph function & inverse on same axes. (switch) x↔y y = x f (x) f –1 (x) x y –2 –1 1 2 –7 9 x y –7 1 2 9 –2 –1

Thought? Do all functions have inverses that are functions? Let’s try one! Or use original & see if it passes horizontal line test! inverse? Not a function – fails vertical line test

So, often we can restrict the domain of a function to make it one-to-one and ensure it has an inverse function Ex 3) Restrict domain to make it one-to-one. Find inverse. State domain & range of inverse function. inverse (switch) *Notice how domain of g is range of g–1 & range of g is domain of g–1