The Lesson
An inverse function is a function that reverses another function. If a function f relates an input x to an output f(x)......an inverse function f^{−1} relates the output f(x) back to the input x:
Imagine a function f relates an input 2 to an output 3...
f(2) = 3
...the inverse function f^{−1} relates 3 back to 2...
f^{−1}(3) = 2
An inverse function is denoted f^{−1}(x).A Real Example of an Inverse Function
It is easier to understand inverse functions with an example. Let us consider a function:f(x) = x + 1 and Its Inverse
The function f(x) = x + 1 takes each input and adds 1 to it. The mapping diagram below shows this function.Now let's consider the inverse function of f(x) = x + 1. The inverse function will take us back to the original values. The inverse function takes each input and subtracts 1 from it. The inverse of f(x) = x + 1 is:
The Inverse Function Takes Us Back to the Input
We can combine a function with its inverse.
Imagine we put a value into a function and then put the result into the inverse function. The output is original value.
If we put an input x into a function f(x), and then put the result into the inverse function f^{−1}(x), the output is x.

Imagine we put a value into the inverse function and then put the result into the function. The output is original value.
If we put an input x into a function f(x), and then put the result into the inverse function f^{−1}(x), the output is x.
Graphs of Inverse Functions
Functions and their inverses can be plotted on graphs. Functions relate an input x to an output y. We can plot these pairs of inputs and outputs as Cartesian coordinates, (x, y).
 The inverse function relates the output y back to the input x. We swap the coordinates around to find the inverse function, such that (x, y) becomes (y, x).