We can think of the *inverse function* of a function as the function, denoted that does the opposite operations that does. In other words, undoes what does to the independent variable (usually ).

More formally, a function *inverse* of a function if for all values in the domain of and for all values in the domain of . If all of that is true, then , and .

We can find the inverse of a function

graphically by reflecting the graph of in the line . This is because we can find the inverse of a function by switching its - and -coordinates.We call a function an *invertible function* if it has an inverse function. We can figure out if a function is invertible by performing the horizontal line test or by reflecting its graph in the line and then seeing if this reflection passes the vertical line test.

In order to figure out if a function has an inverse, we can perform the *horizontal line test* on that function. This is very similar to the vertical line test, except that it uses a horizontal line instead.

To perform the horizontal line test, first graph the function you’re interested in learning about. Then position something straight, like a ruler or a pencil, parallel to the

-axis on the coordinate plane, and move this “horizontal line” up and down along the -axis to see if it intersects the graph of the function more than once at any given position. If it does, then the function is not invertible (i.e. does not have an inverse function). The function passes the horizontal line test, so it has an inverse.The function

does not pass the horizontal line test, as we can see from the graph below, so it is not invertible.