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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:


You can write all of this in one step as:


If you write this in one step, you get:


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


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


These functions are graphed below:


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.

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