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How to Recognize Inverse Functions

You can tell that two functions are inverse functions when each one undoes what the other does. When you graph inverse functions, each is a mirror image of the other. Here are some examples of inverse functions:

image0.png

You can write all of this in one step as:

image1.png

If you write this in one step, you get:

image2.png

Don’t confuse the superscript –1 in a function with exponent –1.

image3.png

When you graph inverse functions, each is the mirror image of the other, reflected over the line
y = x.

image4.png

These functions are graphed below:

image5.jpg

If you rotate the graph in the figure counterclockwise so that the line y = x is vertical, you can easily see that these functions are mirror images of each other. One consequence of this symmetry is that if a point like (2, 4) is on one of the functions, then the point (4, 2) will be on the other.

image6.png
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