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Inverse Functions (College Board AP® Precalculus): Study Guide

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 13 April 2026

Inverse functions

What is an inverse function?

An inverse function is a function that reverses the mapping of an original function
- For the function $f$ , the inverse function is notated by $f^{- 1}$

Examiner Tips and Tricks

Be careful with the notation here!

The notation $f^{- 1} (x)$ means the inverse of $f (x)$
It does not mean $\frac{1}{f (x)}$

If the original function takes input $a$ and produces output $b$
- then the inverse function takes $b$ as input and returns output $a$
  - I.e. if $f (a) = b$
    - then $f^{- 1} (b) = a$
You can think of this in terms of input-output pairs
- If $f$ contains the pair $(a, b)$
  - then $f^{- 1}$ contains the pair $(b, a)$
This means you can find values of an inverse function
- by reading the original function backwards
From a table
- If you know that $f (3) = 7$ , then $f^{- 1} (7) = 3$
  - Look for 7 in the output row, then read the corresponding input
From a graph
- If the point $(3, 7)$ is on the graph of $f$ , then $f^{- 1} (7) = 3$
  - Find 7 on the $y$ -axis, read across to the curve, and then down to the $x$ -axis

When does a function have an inverse?

A function $f$ has an inverse function (i.e. it is invertible) on a given domain
- if every output value comes from exactly one input value
  - In other words, if no two different inputs produce the same output
- Such a function, where no two different inputs produce the same output, is often referred to as a one-to-one function
  - A function must be one-to-one to be invertible
This invertibility requirement comes because an inverse function must be a function
- And a function can only give one output for each input
If a function is not invertible on its full domain, the domain can sometimes be restricted to a smaller interval where it is invertible
- There is often more than one way to restrict the domain to achieve this
E.g. the function $f (x) = x^{2}$ is not invertible on all real numbers
- because both $f (2) = 4$ and $f (- 2) = 4$
  - Two different inputs give the same output
But if the domain is restricted to $x \geq 0$
- then each output comes from a unique input, and the inverse exists
  - In that case $f^{- 1} (x) = \sqrt{x}$

How do you determine if a function is invertible from a table or graph?

From a table
- Check whether any output value appears more than once
  - If it does, the function is not invertible (on the given domain)
  - because multiple inputs map to the same output
From a graph
- You can use the 'horizontal line test'
- Check whether any horizontal line crosses the graph more than once
  - If it does, there are multiple inputs producing the same output,
  - so the function is not invertible

Left graph shows an invertible function; right graph shows a non-invertible function due to the same y-values coming from multiple x-values. — **The horizontal line test**

If the function is always increasing or always decreasing on its domain
- then it is automatically invertible
  - because outputs always changing in only one direction
  - means no two inputs can share the same output

Examiner Tips and Tricks

An exam question may ask you to determine whether a function has an inverse and to justify your answer. Simply stating "the function is not one-to-one" or "it fails the horizontal line test" is not enough to earn the reasoning point. You must refer to specific values from the table or graph.

E.g. " $f$ does not have an inverse function because $f (- 3) = f (0) = f (3) = 1$ , so the output 1 comes from more than one input"

Worked Example

The function $f$ is defined for all real numbers. The table gives values of $f (x)$ at selected values of $x$ .

$x$	$- 4$	$- 2$	0	2	4	6
$f (x)$	10	5	3	5	10	18

The function $g$ is an increasing function defined for all real numbers. The table gives values of $g (x)$ at selected values of $x$ .

$x$	$- 1$	0	1	2	3	4
$g (x)$	$- 5$	$- 2$	0	3	8	15

(a) Find the value of $g^{- 1} (3)$ , or indicate that it is not defined.

Answer:

Because $g$ is increasing, it is invertible

therefore $g^{- 1}$ exists

From the table, $g (2) = 3$ , so

$g^{- 1} (3) = 2$

(b) (i) Determine whether $f$ has an inverse function on its given domain.

(ii) Give a reason for your answer based on the definition of a function and the table of values for $f$ .

Answer:

The output values 5 and 10 each appear more than once in the table for $f$

This means that the function has more than one input producing the same output
therefore it is not invertible

(i)

$f$ does not have an inverse function on its given domain

(ii)

There are output values of $f$ that correspond to more than one input value. For example, $f (- 4) = 10$ and $f (4) = 10$ , so the output value 10 is not mapped from a unique input. Similarly, $f (- 2) = 5$ and $f (2) = 5$ . Because multiple inputs produce the same output, the reverse mapping would not be a function, so $f$ is not invertible.

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Inverse Functions (College Board AP® Precalculus): Study Guide

Inverse functions

What is an inverse function?

Examiner Tips and Tricks

When does a function have an inverse?

How do you determine if a function is invertible from a table or graph?

Examiner Tips and Tricks

Worked Example

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Mark Curtis