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Finding the Inverse of a Function Using a Graph

(KS4, Year 10)

A function and its inverse function can be plotted on a graph. If the function is plotted as y = f(x), we can reflect it in the line y = x to plot the inverse function y = f⁻¹(x). find_inverse_graph

Every point on a function with Cartesian coordinates (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. find inverse function graph example

Step-by-Step:

1

Plot the function on a graph.

find inverse function graph step 1

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 y = x on the same graph.

find inverse function graph step 2

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

3

Reflect the line 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.

find inverse function graph step 3

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.

The equation of the line is:

y = slope x + y-intercept

y = 1x + −1

Answer:

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

Lesson Slides

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

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)...

function x to f(x)

...an inverse function f⁻¹ relates the output f(x) back to the input x:

inverse function f(x)_to_x

Imagine a function f relates an input 2 to an output 3...

f(2) = 3

...the inverse function f⁻¹ relates 3 back to 2...

f⁻¹(3) = 2

An inverse function is denoted 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: mirror_image

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: reflected_function