# Inverse Functions

How to solve inverse functions problems: definition, formula, graphs, relationship between the horizontal line test, examples, and their solutions.

## Definition

Here's *y* = *f*(*x*).

If you change this into [*x* = ...] form,

(with respect to *x*)

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

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

For *y* = *f*(*x*),

if you put *x* in *f*,

you get *y*.

For *x* = *f*^{-1}(*y*),

if you put *y* in *f*^{-1},

you get *x*.

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

## Example 1

*f*(2) = 5 means

if you put 2 in *f*(*x*),

you get 5.

So, if you put 5 into *f*^{-1}(*y*),

you get 2.

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

## Example 2

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

(7 is the *y* value.)

Then *f*(*x*) = 7.

It says *f*(*x*) = 2*x* + 1.

And *f*(*x*) = 7.

So 2*x* + 1 = 7.

Move +1 to the right side.

Then 2*x* = 6.

Divide both sides by 2.

Then *x* = 3.

*x* = 3*f*^{-1}(7) = *x*

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

## Example 3

Set *f*(*x*) = *y*.

It says *f*(*x*) = 2*x* + 4.

So 2*x* + 4 = *y*.

Write this in [*x* = ...] form.

Move +4 to the right side.

Then 2*x* = *y* - 4.

Divide both sides by 2.

Then *x* = (1/2)*y* - 2.

This function is in [*x* = ...] form.

So set the right side as *f*^{-1}(*y*).

*x* = (1/2)*y* - 2

= *f*^{-1}(*y*)

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

It says to find *f*^{-1}(*x*), not *f*^{-1}(*y*).

So change the variable to *y*.

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

## Formula: (*f*^{-1} ∘ *f*), (*f* ∘ *f*^{-1})

The composition of a function and its inverse

returns the original input: *x*.

So (*f*^{-1} ∘ *f*)(*x*) = *x*.

The order of *f* and *f*^{-1} doesn't matter.

So (*f* ∘ *f*^{-1})(*x*) = *x*.

Composite functions

## Example 4

(*f*^{-1} ∘ *f*)(*x*) = *x*

So (*f*^{-1} ∘ *f*)(5) = 5.

You don't have to use *f*(*x*) = *x*^{2} - *x* + 1.

## Example 5

(*f* ∘ *f*^{-1})(*x*) = *x*

So (*f* ∘ *f*^{-1})(7) = 7.

## Graph: *y* = *f*(*x*) and *y* = *f*^{-1}(*x*)

*y* = *f*(*x*) and *y* = *f*^{-1}(*x*)

are symmetric about the line *y* = *x*.

Reason:*y* = *f*(*x*) is *x* = *f*^{-1}(*y*).

Then *y* = *f*^{-1}(*x*) is the image of *x* = *f*^{-1}(*y*)

under the reflection under the line *y* = *x*,

because it shows (*x*, *y*) → (*y*, *x*).

Reflection in the line *y* = *x*

## Relationship between the Horizontal Line Test

*y* = *f*(*x*) and *y* = *f*^{-1}(*x*)

are symmetric about the line *y* = *x*.

So the horizontal line test for *y* = *f*(*x*)

has the same meaning as

the vertical line test for *y* = *f*^{-1}(*x*).

So, to see if *y* = *f*^{-1}(*x*) is an existing function,

do the horizontal test for *y* = *f*(*x*).

If *y* = *f*(*x*) passes the horizontal line test,

then *y* = *f*(*x*) is one-to-one,

and *y* = *f*(*x*) has an inverse function: *f*^{-1}(*x*).

Horizontal line test

Vertical line test

## Example 6

Starting from the bottom,

do the horizontal line test.

Every horizontal line shows one input.

So the function passes the horizontal line test.

Then the function is one-to-one.

The given function *y* = *f*(*x*)

passed the horizontal line test.

So its inverse function *y* = *f*^{-1}(*x*)

passed the vertical line test.

So *y* = *f*^{-1}(*x*) is an existing function.

## Example 7

At the origin,

the horizontal line shows one input.

So, at the origin,

the function passes the horizontal line test.

But if you make a horizontal line like this,

it shows two inputs.

So the graph fails the horizontal line test.

So this function is not one-to-one.

The given function *y* = *f*(*x*)

failed the horizontal line test.

So its inverse function *y* = *f*^{-1}(*x*)

failed the vertical line test.

So *y* = *f*^{-1}(*x*) is not an existing function.