Inverse Function Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Inverse Function.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The inverse of a function $f$ is a function $f^{-1}$ that reverses $f$ : if $f(a) = b$ then $f^{-1}(b) = a$ . It exists only when $f$ is one-to-one.

If $f$ turns $a$ into $b$ , then $f^{-1}$ turns $b$ back into $a$ . Reverse the process.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: An inverse function reverses a function, turning each output back into the input that produced it.

Common stuck point: The procedure for inverse function is the easy part; the trap is reading $f^{-1}$ as a reciprocal. Asking "If I know the output, does this rule hand back the exact input that produced it?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: If I know the output, does this rule hand back the exact input that produced it?

Worked Examples

Example 1

easy

Find the inverse of

f(x) = 3x - 7

Answer

f^{-1}(x) = \frac{x + 7}{3}

First step

Write

y = 3x - 7

Full solution

2
Swap $x$ and $y$ : $x = 3y - 7$ .
3
Solve for $y$ : $3y = x + 7$ , so $y = \frac{x + 7}{3}$ .
4
Therefore $f^{-1}(x) = \frac{x + 7}{3}$ .

To find an inverse, swap input and output then solve for the new output. The inverse 'undoes' the original function: applying

f

then

f^{-1}

returns the original input.

Example 2

medium

Find the inverse of

f(x) = \frac{2x + 5}{x - 1}

for

x \neq 1

Example 3

medium

Find the inverse of

f(x) = \frac{3x - 4}{x + 2}

and state its domain.

Example 4

medium

Find the inverse of

f(x) = \frac{2}{x+3}

and state its domain and range.

Example 5

hard

f(x) = \frac{ax + b}{cx + d}

with

ad - bc \neq 0

, find

f^{-1}(x)

Example 6

hard

For

f(x) = \frac{x}{x+1}

\mathbb{R} \setminus \{-1\}

, find

f^{-1}

and its domain.

Example 7

challenge

For

f(x) = x + \sin x

, prove that

f

is invertible on

\mathbb{R}

without finding

f^{-1}

explicitly.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find the inverse of

f(x) = \frac{x}{4} + 2

Example 2

hard

f(x) = 2^x

, find

f^{-1}(x)

. Then verify by showing that

f(f^{-1}(8)) = 8

Example 3

easy

Find the inverse of

f(x)=x+5

Example 4

easy

Find the inverse of

f(x)=3x

Example 5

easy

f(2)=7

, what is

f^{-1}(7)

Example 6

easy

Find the inverse of

f(x)=2x-4

Example 7

easy

f(x)=x^2

on all reals invertible?

Example 8

easy

Find the inverse of

f(x)=x^3

Example 9

easy

f^{-1}(x)=x-2

, find

f(x)

Example 10

easy

What is

f(f^{-1}(9))

for any invertible

f

Example 11

medium

Find the inverse of

f(x)=\frac{x+1}{x-2}

Example 12

medium

Find the inverse of

f(x)=\sqrt{x-3}

for

x\ge 3

Example 13

medium

Find the inverse of

f(x)=2^x

Example 14

medium

For

f(x)=4x-1

, solve

f^{-1}(x)=2

Example 15

medium

Find the inverse of

f(x)=\frac{1}{x}

Example 16

medium

f

and

g

are inverses and

g(5)=3

, find

f(3)

Example 17

medium

Find the inverse of

f(x)=x^2+1

restricted to

x\ge 0

Example 18

challenge

f(x)=\frac{2x-3}{x+1}

, find

f^{-1}(1)

Example 19

challenge

A linear function satisfies

f(x)=f^{-1}(x)

for all

x

and

f(x)=ax+b

with

a\neq 1

. Find

a

Example 20

challenge

f(g(x))=x

for all

x

and

f(x)=3x+2

, find

g(x)

Example 21

medium

Find the inverse of

f(x)=\frac{x}{2}+3

Example 22

medium

f(x)=\log_3 x

, find

f^{-1}(x)

Example 23

easy

Find the inverse of

f(x) = 5x - 2

Example 24

easy

Find the inverse of

f(x) = \frac{x-7}{2}

Example 25

easy

What is the inverse of

f(x) = x

Example 26

easy

f(x) = |x|

invertible on all real numbers?

Example 27

medium

Find the inverse of

f(x) = \sqrt{x+5}

for

x \geq -5

Example 28

medium

f(x) = e^{3x}

, find

f^{-1}(x)

Example 29

medium

Find

f^{-1}(x)

for

f(x) = \ln(x - 2)

Example 30

medium

For

f(x) = 6 - 2x

, solve

f^{-1}(x) = 4

Example 31

medium

f

is invertible and

f(a) = b

f(b) = c

, what is

f^{-1}(c)

Example 32

medium

On what domain is

f(x) = x^2

invertible, and what is the inverse?

Example 33

medium

Two functions are graphed:

y = f(x)

and

y = f^{-1}(x)

. What is the relationship between the two graphs?

Example 34

medium

For

f(x) = 7

(a constant function), is

f

invertible?

Example 35

hard

Find all linear functions

f(x) = ax + b

that satisfy

f = f^{-1}

Example 36

hard

f(x) = x^3 + 1

, find

(f^{-1})'(2)

. (Use the rule

(f^{-1})'(b) = 1/f'(a)

where

f(a)=b

Example 37

hard

Show that if

f

and

g

are both invertible, then

(f \circ g)^{-1} = g^{-1} \circ f^{-1}

Example 38

hard

Let

f(x) = \tan x

(-\pi/2, \pi/2)

. Find

f^{-1}(1)

and state its general form.

Example 39

hard

Suppose

f: \mathbb{R} \to \mathbb{R}

is invertible and continuous. What can you say about the monotonicity of

f

Example 40

challenge

Find a function

f

that is its own inverse (i.e.,

f \circ f = \mathrm{id}

) and is NOT linear.

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Related Concepts

Function One-to-One Mapping Horizontal Line Test

Background Knowledge

These ideas may be useful before you work through the harder examples.

function definition