An inverse function undoes the operations performed on variables of a function. For example, if a number is multiplied by 2, and then 3 is added, we can write this as the function . The inverse to this function first subtracts 3 from the input value, and then divides by 2, so as to completely undo all operations of the original function. We write this as:
If a function becomes the input of an inverse function, then the output is the argument of the original function. Mathematically, we write this as:
Evaluating an Inverse Graphically
On a graph, x– and y– coordinates between a function and its inverse inverted, or swapped. This means that for any coordinate, (x, y), of a function, we can find a corresponding coordinate on the graph of its inverse using the coordinates (y, x). This means we locate the x–value on the y–axis, and locate the y–value on the x–axis. This is shown in the graph below:
Finding the Inverse Algebraically
If we want to find the inverse of a function algebraically, there are two common procedures most people use:
Let's define the inverse of using one of the methods above:
Technically, we need to restrict the domain of the inverse function to non-negative values of x. This is because the range of the original function was restricted to non-negative y–values. The domain and range of a function also swap when defining the domain and range of its inverse.