**y = f(x)**, we can reflect it in the line

**y = x**to plot the inverse function

**y = f**.Every point on a function with Cartesian coordinates

^{−1}(x)**(x, y)**becomes the point

**(y, x)**on the inverse function: the coordinates are swapped around.

## How to Find the Inverse of a Function Using a Graph

Finding the inverse of a function using a graph is easy.## Question

Find the inverse function of the function plotted below.## Step-by-Step:

## 1

Plot the function on a graph.

## How to Plot a Function

The function is a linear equation and appears as a straight line on a graph.- The constant term gives the y-intercept
In our example, the y-intercept is
**1**. The line crosses the y-axis at 1.

- The coefficient of the
**x**term gives the slope of the line. In our example, there is no number written in front of the**x**. It has an implicit coefficient of**1**. The line has a slope of**1**. The line will go up by 1 when it goes across by 1.

## 2

Plot the line

The line

**y = x**on the same graph.The line

**y = x**is a 45° line, halfway between the x-axis and the y-axis.## 3

Reflect the line

The reflected line is the graph of the inverse function.

**y = f(x)**in the line**y = x**. Each point on the reflected line is the same perpendicular distance from the line**y = x**as the original line.The reflected line is the graph of the inverse function.

## 4

Find the equation of the inverse function.

- The y-intercept is
**−1**.

- The slope is
**1**.

y = **slope** x + **y-intercept**

y = **1**x + **−1**

## Answer:

The inverse of the function**f(x) = x + 1**is:

**f**.

^{−1}(x) = x − 1## What Is an Inverse Function?

An inverse function is a function that reverses another function. If a function**f**relates an input

**x**to an output

**f(x)**...

...an inverse function

**f**relates the output

^{−1}**f(x)**back to the input

**x**:

Imagine a function

**f**relates an input

**2**to an output

**3**...

f(2) = 3

...the inverse function **f**relates

^{−1}**3**back to

**2**...

f

An inverse function is denoted ^{−1}(3) = 2**f**.

^{−1}(x)## How To Reflect a Function in y = x

To find the inverse of a function using a graph, the function needs to be reflected in the line**y = x**. By reflection, think of the reflection you would see in a mirror or in water: Each point in the image (the reflection) is the same perpendicular distance from the mirror line as the corresponding point in the object. If a function is reflecting the the line

**y = x**, each point on the reflected line is the same perpendicular distance from the mirror line as the original function:

## You might also like...

#### Help Us Improve Mathematics Monster

- Do you disagree with something on this page?
- Did you spot a typo?

__this form__.

#### Find Us Quicker!

- When using a search engine (e.g., Google, Bing), you will find Mathematics Monster quicker if you add
**#mm**to your search term.

#### Share This Page

If you like Mathematics Monster (or this page in particular), please link to it or share it with others.

If you do, __please tell us__. It helps us a lot!

#### Create a QR Code

Use our handy widget to create a QR code for this page...or any page.