# Domain

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, ${\displaystyle f(x)={\sqrt {x}}}$, 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 ${\displaystyle \operatorname {dom} (f)}$, where f is the function.

More precisely, given a function ${\displaystyle f\colon X\to Y}$, the domain of f is X. Note that 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 subsets of ${\displaystyle \mathbb {R} }$, 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 ${\displaystyle f\colon X\to Y}$, 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 ${\displaystyle f\colon X\to Y}$ to ${\displaystyle A}$, where ${\displaystyle A\subseteq X}$, is written as ${\displaystyle \left.f\right|_{A}\colon A\to Y}$.

## 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.

### Examples

• The function ${\displaystyle f}$ defined by ${\displaystyle f(x)={\frac {1}{x}}}$ cannot be evaluated at 0. Therefore the natural domain of f is ${\displaystyle \mathbb {R} \setminus \{0\}}$.
• In contrast, if ${\displaystyle f}$ is the piecewise function ${\displaystyle f(x)={\begin{cases}1/x&x\not =0\\0&x=0\end{cases}},}$ then ${\displaystyle f}$ is defined for all real numbers, and its natural domain is ${\displaystyle \mathbb {R} }$.
• The function ${\displaystyle x\mapsto {\sqrt {x}}}$ has as its natural domain the non-negative real numbers, which can be denoted by ${\displaystyle \mathbb {R} _{\geq 0}}$, by the interval ${\displaystyle (0,\infty )}$, or by ${\displaystyle \{x\in \mathbb {R} :x\geq 0\}}$.
• The tangent function ${\displaystyle \tan x}$ has as its natural domain the set of all real numbers which are not of the form ${\displaystyle {\tfrac {\pi }{2}}+k\pi ,}$ where k is any integer.

## Other uses

The word "domain" is used with other related meanings in some areas of mathematics. In topology, a domain is a connected open set. In real and complex analysis, a domain is an open connected subset of a real or complex vector space. In the study of partial differential equations, a domain is the open connected subset of the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ where a problem is posed (i.e., where the unknown function(s) are defined).

## 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.