Given a function , the inverse of is the function which has the property that exactly when (for the same values of and ). That is, the inverse of a function exactly undoes whatever the function does. The inverse of the function is commonly denoted by .
Some functions have inverses (they are called invertible) and some do not (they are called non-invertible). An easy way to tell if a function is invertible or not is whether or not it passes the horizontal line test:
If any horizontal line passes through more than one point on the graph of y=f(x), the function f(x) is not invertible.
If no such horizontal line exists, the function is invertible.
Even if is not invertible, it might still have a partial inverse. If you restrict the domain of , creating a new function which does pass the horizontal line test, then is invertible, and its inverse is called a partial inverse of .
In order to graph a function's inverse, simply reflect its graph through the line .
Note : Even though they look similar, the inverse of , denoted , is not the same as the reciprocal of , which can be written . For example, the inverse of the function is the function , not the expression . The notation is intended to represent the concept of "inverting the action of ", not "inverting the result of ".