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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-1(x) is found.



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.
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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-1(x). It relates an input 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: