# How to Find the Inverse of a Function Using a Graph (Mathematics Lesson)

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

A function can be plotted on a graph.If the graph of

**y = f(x)**is plotted, and then reflected in the line

**y = x**, the inverse function

**y = f**is found.

^{-1}(x)# A Real Example of How to Find the Inverse of a Function Using a Number Machine

**Question:**What is the inverse function of the function:

**Step 1:**Plot the function on a graph.

The function is a linear equation, and appears as a straight line on a graph.

The slope of the line is given by the of the

**x**term, 1. The y-intercept is given by the constant term

**+ 1**.

Plotting the function gives:

**Step 2:**Plot the line

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

The line

**y = x**goes through the origin and is exactly halfway between the y-axis and the x-axis, at a 45° angle to them:

**Step 3:**Reflect the function

**y = f(x)**in the line

**y = x**.

**Step 4:**Find the equation of the reflected line, which is the inverse function.

The inverse function is a straight line. It has the same slope as the original function,

**1**, but its y-intercept is

**-1**:

The inverse function is:

# Another Real Example of How to Find the Inverse of a Function Using a Graph

The slider below shows a real example of how to find the inverse of a function using a graph.##### Interactive Test

**show**

##### Note

**WHAT IS AN INVERSE FUNCTION?**

An inverse function is itself a function which reverses a function.

If a function

**f(x)**maps an input

**x**to an output

**f(x)**...

... an inverse function takes the output

**f(x)**back to the input

**x**:

An inverse function is denoted

**f**. It relates an input

^{-1}(x)**x**to an output

**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: