Inverse functions are functions that when combined to form a composite function, they cancel each other out and have an output equal to the original input. Graphically, the inverse function of a function is the reflection of that function across y=x. It can also be found by switching the x and y values of the coordinate points and graphing those points. For example, if a function contained the points (1,4), (2,5), and (3,6), it’s inverse function would contain the points (4,1), (5,2), and (6,3). This method may be hard to execute for more complex function equations, so the algebraic method of finding the inverse is used. The function we will find the inverse of is f(x)=2x+4, with a domain of 1<x<5. This domain produces the range of 6<y<14. The first step is switching the input and output variables so the function reads x = 2f(x)+4. The next step is to then solve for f(x). First, 4 would be subtracted from x, then that value would be divided by 2, so f(x) = (x-4)/2. The domain and range are then reversed to give a domain of 6<x<14 and range of 1<y<5. This is possible because of the reflection across y=x. When inverse functions are combined to form a composite function, the expression for the function is just x, or the input.
However, sometimes the inverse of a function is not a function. An example of this kind of function would be a parabola. When a parabola is reflected across y=x, it creates a horizontal parabola. As explained in the Functions section, a horizontal parabola is not a function because it has 2 outputs for a single input. Some non-functions inverses are in fact functions. If we use a horizontal parabola as the original function, it’s inverse would be a vertical parabola, which is a function.