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In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by $\mathrm{dom}\left(f\right)$ or $\mathrm{dom}f$, where f is the function. In layman's terms, the domain of a function can generally be thought of as "what x can be".^[1]

More precisely, given a function $f:X\to Y$, the domain of f is X. In modern mathematical language, the domain is part of the definition of a function rather than a property of it.

In the special case that X and Y are both sets of real numbers, the function f can be graphed in the Cartesian coordinate system. In this case, the domain is represented on the x-axis of the graph, as the projection of the graph of the function onto the x-axis.

For a function $f:X\to Y$, the set Y is called the codomain: the set to which all outputs must belong. The set of specific outputs the function assigns to elements of X is called its range or image. The image of $f$ is a subset of Y, shown as the yellow oval in the accompanying diagram.

Any function can be restricted to a subset of its domain. The restriction of $f:X\to Y$ to $A$, where $A\subseteq X$, is written as ${f|}_A:A\to Y$.

If a real function f is given by a formula, it may be not defined for some values of the variable. In this case, it is a partial function, and the set of real numbers on which the formula can be evaluated to a real number is called the natural domain or domain of definition of f. In many contexts, a partial function is called simply a function, and its natural domain is called simply its domain.

The term domain is also commonly used in a different sense in mathematical analysis: a domain is a non-empty connected open set in a topological space. In particular, in real and complex analysis, a domain is a non-empty connected open subset of the real coordinate space ${ℝ}^n$ or the complex coordinate space ${ℂ}^n.$

Sometimes such a domain is used as the domain of a function, although functions may be defined on more general sets. The two concepts are sometimes conflated as in, for example, the study of partial differential equations: in that case, a domain is the open connected subset of ${ℝ}^n$ where a problem is posed, making it both an analysis-style domain and also the domain of the unknown function(s) sought.

For example, it is sometimes convenient in set theory to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple (X, Y, G). With such a definition, functions do not have a domain, although some authors still use it informally after introducing a function in the form f: X → Y.^[2]

• Argument of a function
• Attribute domain
• Bijection, injection and surjection
• Codomain
• Domain decomposition
• Effective domain
• Endofunction
• Image (mathematics)
• Lipschitz domain
• Naive set theory
• Range of a function
• Support (mathematics)

• Bourbaki, Nicolas (1970). Théorie des ensembles. Éléments de mathématique. Springer. ISBN 9783540340348.
• Eccles, Peter J. (11 December 1997). An Introduction to Mathematical Reasoning: Numbers, Sets and Functions. Cambridge University Press. ISBN 978-0-521-59718-0.
• Mac Lane, Saunders (25 September 1998). Categories for the Working Mathematician. Springer Science & Business Media. ISBN 978-0-387-98403-2.
• Scott, Dana S.; Jech, Thomas J. (31 December 1971). Axiomatic Set Theory, Part 1. American Mathematical Soc. ISBN 978-0-8218-0245-8.
• Sharma, A. K. (2010). Introduction To Set Theory. Discovery Publishing House. ISBN 978-81-7141-877-0.
• Stewart, Ian; Tall, David (1977). The Foundations of Mathematics. Oxford University Press. ISBN 978-0-19-853165-4.