Inverse Functions

Calculus

Section 1.3 Things

Inverse Functions

Suppose we had a function where \(f(-7)=11\).   Is it possible to start with \(x=11\) and get back \(-7\)?   When this is possible, it is called the inverse function.   This brings two questions:   When is this possible? and What is the inverse function?

When does an inverse function exist?

The inverse has to be a function as well to avoid something like \(f(5)=12\) and \(f(-5)=12\).   This would lead to the inverse having \(x=12\) for \(y=\pm5\), and that violates the definition of a function!!   To avoid this scenerio, an inverse will exist when each \(x\)-value has EXACTLY one \(y\)-value.   This is called a one to one function, and there is an easy way to test for it.   Consider the following graphs:

cubic root
Not one to one One to one
log abs
One to one Not one to one

Did you figure out what is going on yet?   It is called the Horizontal Line Test.   A function is one to one if it passes the Horizontal Line Test, so an inverse then exists.

How do we find the inverse function?

Finding an inverse is a very procedural thing to do, and may require some high powered algebra to complete.   Below is the step by step procedure:

Step 1: Write function in \(y=\) form
Step 2: Switch the \(x\) with every \(y\)
Step 3: Algebraically solve the equation for \(y\)
Step 4: Replace \(y\) with \(f^{-1}\)

Example   Find the inverse of \(f(x)=(x+5)^3-4\).