Helpful Tools for Calculus, Chapter 1: Functions and Relations
Previous Section: 1.4 Polynomial and Rational Functions
Functions and their Inverse
A function is said to have an inverse if there exists another function such that their composition results in the variable itself. This sounds confusing but think of it in terms of operations first:
An inverse operation is one that cancels out another operation. Addition and subtraction are inverses, just as multiplication and division are. An inverse function is a function that cancels out another function. On a graph, inverses can be identified because they are reflections of each other over the diagonal line, y=x.
If we take our previous examples, we can rewrite them like this:
In this case, f(x) and g(x) are inverse functions. It is not the case that a function has an inverse! A function that has an inverse is called invertible. The notation for an inverse is somewhat confusing. To denote an inverse we write f -1(x). Be careful! This does not mean the same thing as the exponent -1. Again: 
, unless it is an involution such as f(x)=x or f(x)=1/x.
To find the inverse of a function, if it exists, we can follow three easy steps:
1. Explicitly state the function in terms of its output, y
2. Switch the x’s and y’s
3. Solve for the ‘new’ y value. This represents the inverse.
Example 1: Find the inverse of f(x)=4x2+3
Solution:
Example 2: Find the inverse of 
Solution:
Domain and Range
The domain of an inverse function is the range of the original function. Similarly, the range of the inverse function is the domain of the original function. If the original function is unknown, you could identify the domain using any of the previous methods.
Example 3: Identify the domain and range of the inverse of 
Solution: We know that that domain of f(x) is x≥3 and the range is y≥0. So the inverse will have these values switched. f -1(x) will have a domain of x≥0 and a range of y≥3.
Example 4: Identify the inverse of and write its domain and range f(x)=-2x3+9
Solution:
As we will talk about in the next section, a cube root has no domain restrictions and can have positive and negative answers. So, the domain and range are x,y∈
.
Review
An Inverse Function is a function that ‘cancels’ out another function. Not all functions have inverses.
The Domain of an Inverse Function is equivalent to the Range of its Inverse.
The Range of an Inverse Function is equivalent to the Domain of its Inverse.
An Inverse Function can be notated as f -1(x).
It is not always the case that
To find and Inverse, switch the x and y values and solve for the new ‘y‘.