# Inverse Function

How to find the inverse function: definition, property, 4 examples, and their solutions.

## Definition

### Definition

Here's y = f(x).

If you change this to [x = ...]

(with respect to x),

then x = f^{-1}(y).

This f^{-1} is the inverse function of f.

## Example 1

### Example

### Solution

f(2) = 5

Then f^{-1}(5) = 2.

So f^{-1}(5) = 2.

## Example 2

### Example

### Solution

Set f^{-1}(7) = x.

f^{-1}(7) = x

Then f(x) = 7.

f(x) = 2x + 1

f(x) = 7

So 2x + 1 = 7.

Move +1 to the right side.

Then 2x = 6.

Divide both sides by 2.

Then x = 3.

f^{-1}(7) = x

x = 3

So f^{-1}(7) = 3.

[∴] means [therefore].

So 3 is the answer.

## Example 3

### Example

### Solution

Set 2x + 4 = y.

Change this to [x = ...].

Move +4 to the right side.

Then 2x = y - 4.

Divide both sides by 2.

Then x = [1/2]y - 2.

x = [1/2]y - 2

So f^{-1}(y) = [1/2]y - 2.

f^{-1}(y) = [1/2]y - 2

Change y to x.

Then f^{-1}(x) = [1/2]x - 2.

So f^{-1}(x) = [1/2]x - 2.

## Property

### Property

The composite function

of a function and its inverse is x.

(f^{-1} ∘ f)(x) = x

(f ∘ f^{-1})(x) = x

y = f(x)

So (f^{-1} ∘ f)(x)

= f^{-1}(f(x))

= f^{-1}(y)

= x.

## Example 4

### Example

### Solution

(f^{-1} ∘ f)(x) = x

So (f^{-1} ∘ f)(3) = 3.

It doesn't matter what f is.

So 3 is the answer.