Finding an Inverse Function

Functions take x values to y values. Inverse functions reverse the direction. Now y values go to x values.

Some equations are easy to find the inverse of others because it is easy to see how to reverse the function for all values.

Here are 3 examples of increasing complexity:

1. If f(x) = x + 3, then we can reverse the function by solving x = f(x) + 3 for f(x), which is f(x) = x – 3.

2. If f(x) = 5x, then we can reverse the function by solving x = 5 f(x) for f(x), which is

3. If f(x)=x³, then we can reverse the function by solving for x = (f(x))³ which is

Same Content from OpenStax

We can now consider one-to-one functions and show how to find their inverses. Recall that a function maps elements in the domain of f to elements in the range of f. The inverse function maps each element from the range of f back to its corresponding element from the domain of f. Therefore, to find the inverse function of a one-to-one function f, given any y in the range of f, we need to determine which x in the domain of f satisfies f(x)=y. Since f is one-to-one, there is exactly one such value x. We can find that value x by solving the equation f(x)=y for x. Doing so, we are able to write x as a function of y where the domain of this function is the range of f and the range of this new function is the domain of f. Consequently, this function is the inverse of f, and we write x=f⁻¹(y). Since we typically use the variable x to denote the independent variable and y to denote the dependent variable, we often interchange the roles of x and y, and write y=f⁻¹(x). Representing the inverse function in this way is also helpful later when we graph a function f and its inverse f⁻¹ on the same axes.

PROBLEM-SOLVING STRATEGY: FINDING AN INVERSE FUNCTION

1. Solve the equation y=f(x) for x.
2. Interchange the variables x and y and write y=f−1(x).