---
title: "What Is the Domain of a Function?"
url: https://www.learningloftinstitute.com/glossary/domain-of-a-function
source: Learning Loft Institute
language: en
updated: 2026-08-21
section: "Glossary › What Is the Domain of a Function?"
---
# What Is the Domain of a Function?
> **Answer.** The domain of a function is the set of inputs it accepts. Unless a question restricts it, the domain is every real number except the ones that break the rule: values that make a denominator zero, and values that put a negative number under a square root.
## Key facts
| | |
| --- | --- |
| Definition | The set of input values a function is allowed to take — every x for which the rule produces a real answer. |
| Where students meet it | Grade 9 or 10 mathematics, once functions are written in f(x) notation, and on the Digital SAT in questions asking which values are not permitted. |
## The two things that rule a value out
At GCSE and IGCSE level, almost every domain question comes down to one of two problems. Either a value would make a denominator zero, or it would ask for the square root of a negative number. Scan the rule for a fraction and a root sign, and if neither is present the domain is usually every real number.
The method is the same in both cases: write down the condition the expression demands, then solve it as an inequality. A denominator gives x − 3 ≠ 0, so x ≠ 3. A root gives x − 2 ≥ 0, so x ≥ 2. The answer to a domain question is an inequality, not a number.
Watch for both appearing together. In 1/√(x − 2) the root demands x ≥ 2 and the denominator forbids the value that makes the root zero, so the domain is x > 2 and the boundary point is excluded after all.
## Domains that are handed to you
Sometimes there is nothing to work out because the question has already decided. f(x) = x² for 0 ≤ x ≤ 5 is a deliberately shortened function, and the restriction is part of its definition rather than a hint about it.
Restricting a domain is not decoration. It changes what the function can output, and it can change what the function is capable of. f(x) = x² over all real numbers cannot be inverted, because 9 would have to come back as both 3 and −3. The same rule restricted to x ≥ 0 can be, and that restriction is why √ has a single agreed value.
Context supplies restrictions too. A function giving the area of a rectangle in terms of one side has a domain limited by the fact that a length cannot be negative, whatever the algebra would tolerate.
## How to write the answer down
Inequality form is the safest: x ≥ 2, or x ≠ 3, or 0 ≤ x ≤ 5. Set notation such as {x : x ≥ 2} appears in some textbooks and means the same thing, and writing all real numbers except 3 in words is accepted in most mark schemes.
The mistake to avoid is answering a different question. The domain is what may go in; the roots are the inputs that produce zero; the range is what comes out. A student who answers x = 3 to a domain question about 1/(x − 3) has found the one value that is definitely not in the domain.
## Questions
### What is the domain of 1/(x − 3)?
Every real number except 3, written x ≠ 3. At x = 3 the denominator becomes zero and the function has no value to return. Every other input, including large negatives and numbers extremely close to 3, works perfectly well — it is a single point removed from the number line, not a region.
### Is the domain of √x written x > 0 or x ≥ 0?
x ≥ 0. Zero is allowed, because √0 = 0 is a real answer like any other. Only negatives are excluded, since no real number squares to give a negative. Losing the equals part of the sign is a common single-mark error, and it appears in almost every root-based domain question.
### What is the difference between domain and range?
Domain is what goes in, range is what comes out. On a graph the domain is measured along the x-axis and the range along the y-axis. Questions often ask for both in one part, and answering them the wrong way round loses both marks even though the working was right.
### Why does the domain matter when finding an inverse?
Because a function can only be inverted if no two inputs share an output. f(x) = x² fails on all real numbers, since 3 and −3 both give 9 and the inverse would not know which to return. Restricting the domain to x ≥ 0 removes the ambiguity and makes the inverse well defined.
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