Inverse Functions
A function and its inverse “un-do,” one another. Their inputs and outputs are switched. For example, if a function has an input of 8 and an output of 11, its inverse would have an input of 11 and an output of 8. That would be the case for every possible input/output pairing!
It’s a little more complicated than that, but that’s the general idea. On this page you’ll learn all about the, “a little more complicated than that,” stuff. Most of it is pretty straight forward, not too mathematically complicated. But, as is the case with functions, the notation is super important. They communication relationship and meaning!
Read through the notes below, watch the video, try the practice problems. Then, you should have a solid understanding of the basics of inverse functions.
A function and its inverse have a special relationship. If a function has an input of A, and an output of B, then the inverse of that function will have an output of A if the input was B. That would work for any input and output combination.
We also know that the notation for an inverse function uses a superscript -1. That is, the -1 looks like an exponent. In the context of functions, it is not a negative exponent if it is written by the functions name. It is the inverse of the function. So the function h-1(x) is the inverse of the function named h(x).
So, for example, consider the function below.
If we were asked to solve the following for the value of x, we could apply what we know to be true about inverse functions.
Here we can see that the output for the inverse function is 2, and we’re asked to find the input. If the output of the inverse is two, then that two is the input for the original function.
This is an important property of inverse functions that should not be forgotten as we dig deeper into inverse functions. In this section we will (1) defined inverse functions, then learn how to (2) find an inverse of a given function, and (3) verify if two functions are inverse.
Definition
Two functions are inverse if their composition both directions results in an output of x.
Let’s look at an example.
These two functions are inverse IF 
That means that if you perform the composition of functions in both directions, the result is x.
Because the composition both directions is x, these are inverse functions. We could then write:
Finding an Inverse Function
If you’re given a function and asked to find its inverse, you can apply a few approaches. In the function is simple, like the examples we used to define inverse functions, you can just figure it out by applying the inverse operation.
For example, if h(x) = 3x, the operation is 3 multiplied by the input. The inverse would be the input divided by 3.
We can verify this is an inverse by performing the composition both directions and having a resulting value of x.
An important thing to remember is that we do the inverse operation, and the inverse order of operations. Let’s see an example.
Here, the order of operations is times 8, then plus 1. If we invert this, we subtract one first, then divide by eight. We need to write this “subtract one first,” as a group to preserve that order. Here’s how:
If things get much more complicated it would be best to have a simple procedure. Here’s what we do, and why.
Finding an Inverse Function
Verification
To verify if two functions are inverse we apply the definition. The definition of inverse functions states the composition of the functions, in both directions, results in an output of x, if the functions are in fact inverse.
Let’s see an example. Verify if the functions below are inverse.
We must perform the composition both directions and see if the result is x, for both. Let’s start.
Now, let’s verify the other direction.
Because the composition of functions in each direction resulted in an output of x, these functions are inverse functions.
Summary:
- A function and its inverse have exchanged inputs and outputs.
- Two functions are inverse if their composition in each direction results in x.
- To find an inverse function:
- Replace the function name with y.
- Exchange x and y.
- Solve for the “new” y.
- Rewrite the “new” y as the inverse function.
- To verify, perform the composition both ways and see if the results for each is just the input of x.
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