How to Find the Inverse of a Function Using a Graph

Finding the Inverse of a Function Using a Graph

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).

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.

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 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 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.

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.

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

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

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See Also

What is a function? What are Cartesian coordinates? What is a linear equation (in slope-intercept form?) Reflecting a shape in y = x using Cartesian coordinates