by Dr. Carol JVF Burns (website creator)
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Before reading this lesson, make sure you fully understand the concepts in the previous two lessons:
using a function box ‘backwards’
one-to-one functions

Often, you need to ‘undo’ what a function did.

For example:

Not all functions can be ‘undone’.
To ‘undo’ means to go from an output back to the input it came fromthis only works if the output came from only one place!
That is, in order to ‘undo’ a function, it must satisfy the following equivalent conditions:

When it exists, the ‘undoing’ function is given a special nameit is called the inverse function.
This lesson makes these ideas precise.

Undoing a One-to-One Function

If $\,f\,$ is one-to-one, then there exists a unique function $\,f^{-1}\,$ (read as ‘$\,f\,$ inverse’) that ‘undoes’ what $\,f\,$ does, as the diagram below illustrates:

  1. start with $\,x\,$
  2. let $\,f\,$ act on $\,x\,$, giving $\,f(x)\,$
  3. rename $\,f(x)\,$ as $\,y\,$
  4. let $\,f^{-1}\,$ act on $\,y\,$, giving $\,f^{-1}(y)\,$
Since this returns us to where we started, $\,f^{-1}(y) = x\,$.

There are two mathematical sentences that emerge in this diagram:

$f(x) = y$ $f^{-1}(y) = x$
‘$\,f\,$ takes $\,x\,$ to $\,y\,$’ ‘$\,f^{-1}\,$ takes $\,y\,$ to $\,x\,$’

These two sentences are equivalentthey are true at the same time, and false at the same time.
This defines the relationship between a function and its inverse:   if one function does something, the other undoes it!

More precisely, the equivalence of these two sentences gives all the following information:

Getting the sentences $\,y = f(x)\,$ and $\,f{\,}^{-1}(y) = x\,$ from each other

going from   $\,y = f(x)\,$   to   $\,f{\,}^{-1}(y) = x\,$  :

let $\,f^{-1}\,$ act on both sides
going from   $\,f{\,}^{-1}(y) = x\,$   to   $\,y = f(x)\,$   :

let $\,f\,$ act on both sides
Start with the equation: $y = f(x)$ Start with the equation: $f^{-1}(y) = x$
Let $\,f^{-1}\,$ act on both sides: $f^{-1}(y) = f^{-1}\bigl(f(x)\bigr)$ Let $\,f\,$ act on both sides: $f\bigl(f^{-1}(y)\bigr) = f(x)$
Since $\,f\,$ and $\,f^{-1}\,$ ‘undo’ each other,
they ‘cancel out’
(this is made precise in the next lesson)
$f^{-1}(y) = \underbrace{f^{-1}\bigl(f(x)\bigr)}_{= x}$ Since $\,f\,$ and $\,f^{-1}\,$ ‘undo’ each other,
they ‘cancel out’
(this is made precise in the next lesson)
$\underbrace{f\bigl(f^{-1}(y)\bigr)}_{= y} = f(x)$
This leaves us with: $f^{-1}(y) = x$ This leaves us with: $y = f(x)$

Caution! $\,f{\,}^{-1}\,$ does NOT mean $\,\frac{1}{f}\,$!

There is some unfortunate notation used for inverse functions, which can lend itself to confusion if you're not careful.

You know from properties of exponents that $\,x^{-1}\,$ denotes the multiplicative inverse of $\,x\,$. That is, $\,x^{-1} = \frac{1}{x}\,$.

However, when $\,f\,$ is a (one-to-one) function, then $\,f^{-1}\,$ does NOT mean $\,\frac{1}{f}\,$!
Instead, $\,f^{-1}\,$ is just notation for the inverse of $\,f\,$the unique function that ‘undoes’ what $\,f\,$ does. Be careful about this!


Let $\,f(x) = x^3\,$.
Answer the following questions:

  1. Is $\,f\,$ one-to-one? Give a reason to support your answer.
    Solution: Yes. The graph of the cubing function passes the horizontal line test.
  2. Does $\,f\,$ have an inverse? Give a reason to support your answer.
    Solution: Yes. Every one-to-one function has an inverse.
  3. Give a formula for $\,f^{-1}(x)\,$.
    Solution: The cube root function undoes the cubing function:   $\,f^{-1}(x) = \root 3\of{x}$
  4. Specialize the equivalent sentences   $\,y = f(x)\,$   and   $\,f^{-1}(y) = x\,$   to the given functions.
    • $\,y = f(x)\,$ becomes $\,y = x^3\,$
    • $f^{-1}(y) = x\,$ becomes $\root 3\of{y} = x$
    Thus, $\,y = x^3\,$ is equivalent to $\,\root 3\of{y} = x\,$.

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
properties of inverse functions
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
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(MAX is 15; there are 15 different problem types.)