How can we use the concept:

To find the inverse of a function, simply interchange the inputs (usually x) and the outputs (usually y) and then solve for the output (y).

For example, find the inverse of f(x) = 2x + 3. Remember the f(x) is pronounced “f of x” and it represents the outputs of the function. So we can rewrite this as y = 2x + 3. When we interchange the inputs and outputs, it looks like x = 2y + 3. Then we solve for y. This requires us to subtract 3 from both sides and then divide by 2. It results in

Now we will put the function notation back. The proper way to note an inverse is with a superscript -1 after the name of the function. In our example, this results in

As mentioned earlier, one of the most exciting things about inverses is their graphs! The graph of our functions from the example looks like this:

Notice that there is a line of symmetry in this graph along the line y = x. In other words, the inverse of the function is a reflection over the line y = x. This is true of all inverses!

Last but not least, it is important to note that not every function has an inverse that is also a function. Remember that a function is a relationship where every input has exactly one output. There are functions where multiple inputs map to the same output and that’s ok; it still satisfies the definition. For example, in the function

the input 2 maps to 4 and the input -2 maps to 4. However, if you find the inverse of this function, that would cause 4 to map to 2 and 4 to map to -2. Now we have a problem, because the same input maps to two different outputs. Therefore, the inverse of this function is not a function.

There is a quick and easy way to tell if the inverse of a function will be a function. Called the Horizontal Line Test, it involves passing a horizontal line over the graph of a function. If the line never hits 2 or more points at the same time, it tells you that the inverse of the function will be a function.