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.