1. 1.6 Inverse Functions

2. 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)}

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

4. 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

5. 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

6. 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