Domain Examples in Math

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

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 domain of a function is the complete set of allowable input values for which the function produces a defined, valid output.

The domain is the list of valid "questions" you can ask the function — values outside the domain produce undefined or meaningless answers.

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: The domain is every input value you are allowed to feed in without breaking the rule.

Common stuck point: The procedure for domain is the easy part; the trap is listing where the function equals zero instead of where it is undefined. Asking "Which input values would make the rule undefined, and have I excluded exactly those?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Which input values would make the rule undefined, and have I excluded exactly those?

Worked Examples

Example 1

easy

Find the domain of

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

Answer

[3, \infty)

First step

The expression under a square root must be non-negative:

x - 3 \geq 0

Full solution

2
Solve: $x \geq 3$ .
3
The domain is $[3, \infty)$ .

Square root functions require non-negative radicands. Setting the expression inside the root

\geq 0

and solving gives the domain.

Example 2

medium

Find the domain of

g(x) = \frac{1}{x^2 - 4}

Example 3

easy

Find the domain of

f(x) = \dfrac{3}{x^2 + 1}

Example 4

medium

Find the domain of

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

Example 5

hard

Find the domain of

f(x) = \dfrac{1}{\ln(x-1)}

Practice Problems

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

Example 1

medium

Find the domain of

h(x) = \frac{\sqrt{x + 1}}{x - 5}

Example 2

hard

Find the domain of

f(x) = \ln(9 - x^2)

Example 3

easy

Find the domain of

f(x)=2x+7

Example 4

easy

Find the domain of

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

Example 5

easy

Find the domain of

f(x)=\sqrt{x-2}

Example 6

easy

Find the domain of

f(x)=\frac{5}{x}

Example 7

easy

Find the domain of

f(x)=x^2-4x+1

Example 8

easy

Find the domain of

f(x)=\ln(x)

Example 9

easy

Find the domain of

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

Example 10

easy

Find the domain of

f(x)=\frac{x+2}{x^2-9}

Example 11

medium

Find the domain of

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

Example 12

medium

Find the domain of

f(x)=\sqrt{9-x^2}

Example 13

medium

Find the domain of

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

Example 14

medium

Find the domain of

f(x)=\log(x-1)+\log(7-x)

Example 15

medium

Find the domain of

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

Example 16

medium

Find the domain of

f(x)=\sqrt{x^2-4}

Example 17

medium

Find the domain of

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

Example 18

challenge

Find the domain of

f(x)=\frac{\sqrt{x+3}}{\sqrt{5-x}}

Example 19

challenge

For which values of

k

is the domain of

f(x)=\frac{1}{x^2+kx+4}

all real numbers?

Example 20

challenge

Find the domain of

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

Example 21

medium

Find the domain of

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

Example 22

medium

Find the domain of

f(x)=\log_2(2x-6)

Example 23

easy

Find the domain of

f(x) = 3x - 5

Example 24

easy

Find the domain of

f(x) = \dfrac{1}{x+4}

Example 25

easy

Find the domain of

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

Example 26

easy

Find the domain of

f(x) = x^3 - x

Example 27

easy

Find the domain of

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

Example 28

medium

Find the domain of

f(x) = \dfrac{\sqrt{x - 3}}{x - 7}

Example 29

medium

Find the domain of

f(x) = \sqrt{16 - x^2}

Example 30

medium

Find the domain of

f(x) = \dfrac{1}{\sqrt{x - 4}}

Example 31

medium

Find the domain of

f(x) = \log(x^2 - 9)

Example 32

medium

Find the domain of

f(x) = \dfrac{x+1}{x^2 - 5x + 6}

Example 33

medium

Find the domain of

f(x) = \tan x

[0, 2\pi]

Example 34

medium

Find the domain of

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

Example 35

medium

Find the domain of

f(x) = \ln(x^2 + 4)

Example 36

hard

Find the domain of

f(x) = \dfrac{\sqrt{x - 1}}{\sqrt{6 - x}}

Example 37

hard

Find the domain of

f(x) = \sqrt{x^2 - 5x + 6}

Example 38

hard

Find the domain of

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

Example 39

hard

For what values of

k

does

f(x) = \sqrt{x^2 + kx + 1}

have domain all reals?

Example 40

hard

Find the domain of

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

Example 41

hard

Find the domain of

f(x) = \ln\!\left(\dfrac{x-3}{x+1}\right)

Example 42

challenge

Find the domain of

f(x) = \sqrt{\sin x}

[0, 2\pi]

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Related Concepts

Function Range Restricted Domain

Background Knowledge

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

function definition