Domain of a function

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Short description: Mathematical concept
A function f from X to Y. The set of points in the red oval X is the domain of f.
Graph of the real-valued square root function, f(x) = x, whose domain consists of all nonnegative real numbers

In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by [math]\displaystyle{ \operatorname{dom}(f) }[/math] or [math]\displaystyle{ \operatorname{dom }f }[/math], 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 [math]\displaystyle{ f\colon X\to Y }[/math], 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 [math]\displaystyle{ f\colon X\to Y }[/math], the set Y is called the codomain, and the set of values attained by the function (which is a subset of Y) is called its range or image.

Any function can be restricted to a subset of its domain. The restriction of [math]\displaystyle{ f \colon X \to Y }[/math] to [math]\displaystyle{ A }[/math], where [math]\displaystyle{ A\subseteq X }[/math], is written as [math]\displaystyle{ \left. f \right|_A \colon A \to Y }[/math].

Natural 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.


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

Other 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 [math]\displaystyle{ \R^n }[/math] or the complex coordinate space [math]\displaystyle{ \C^n. }[/math]

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 [math]\displaystyle{ \R^{n} }[/math] where a problem is posed, making it both an analysis-style domain and also the domain of the unknown function(s) sought.

Set 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: XY.[2]

See also