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Domain of a function

Set of all things that may be the input of a mathematical function

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated August 14, 2026
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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 \operatorname {dom} (f) or \operatorname {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".

More precisely, given a function f\colon 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\colon 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\colon X\to Y to A, where A\subseteq X, is written as \left.f\right|_{A}\colon A\to Y.

01Natural domain

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.

Examples

  • The function f defined by f(x)={\frac {1}{x}} cannot be evaluated at 0. Therefore, the natural domain of f is the set of real numbers excluding 0, which can be denoted by \mathbb {R} \setminus \{0\} or \{x\in \mathbb {R} :x\neq 0\}.
  • The piecewise function f defined by f(x)={\begin{cases}1/x&x\not =0\\0&x=0\end{cases}}, has as its natural domain the set \mathbb {R} of real numbers.
  • The square root function f(x)={\sqrt {x}} has as its natural domain the set of non-negative real numbers, which can be denoted by \mathbb {R} _{\geq 0}, the interval [0,\infty ), or \{x\in \mathbb {R} :x\geq 0\}.
  • The tangent function, denoted \tan, has as its natural domain the set of all real numbers which are not of the form {\tfrac {\pi }{2}}+k\pi for some integer k, which can be written as \mathbb {R} \setminus \{{\tfrac {\pi }{2}}+k\pi :k\in \mathbb {Z} \}.
Graph of the arcsine and arccosine functions, f(x) = arcsin(x) and f(x) = arccos(x), each of whose domain consists of the set of real numbers [-1,1] inclusively
Graph of the arcsine and arccosine functions, f(x) = arcsin(x) and f(x) = arccos(x), each of whose domain consists of the set of real numbers [-1,1] inclusively

02Other uses

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 \mathbb {R} ^{n} or the complex coordinate space \mathbb {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 \mathbb {R} ^{n} where a problem is posed, making it both an analysis-style domain and also the domain of the unknown function(s) sought.

03Set theoretical notions

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.

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Sources and credits

This article is adapted from the Wikipedia article “Domain of a function”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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