# Inverse Function: Graph

How to find out if the given function has an inverse function: graph, 2 examples, and their solutions.

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

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

show the reflection in y = x.

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

Inverse Function

Then see x = f^{-1}(y) and y = f^{-1}(x).

x and y are switched.

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

show the reflection in y = x.

## Horizontal Line Test

So doing the horizontal line test

for y = f(x) is

doing the vertical line test

for its inverse y = f^{-1}(x).

So, if y = f(x) passes the horizontal line test,

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

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

This means

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

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

## Example

To see if y = f(x) has an inverse function,

do the horizontal line test.

From the bottom side of the graph,

think of a horizontal line.

Move the horizontal line to the top side.

And see if the horizontal line and the graph

intersect at [one point].

As you can see,

the line and the graph intersect at one point.

So y = f(x) passes the horizontal line test.

So y = f(x) has an inverse function.

## Example

To see if y = f(x) has an inverse function,

do the horizontal line test.

If the horizontal line is like this,

the line and the graph intersect at one point.

So, for this case,

y = f(x) passes the test.

But if the horizontal line is like this,

the line and the graph intersect at two points:

not at one point.

So y = f(x) fails the horizontal line test.

So y = f(x) does not have an inverse function.