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A function $f$ from $X$ to $Y$ . The set of points in the red oval $X$ is the domain of $f$ .

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".^[1]

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 $\left.f\right|_{A}\colon A\to Y$ .

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 $f$ 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 $\mathbb {R} \setminus \{{\tfrac {\pi }{2}}+k\pi :k\in \mathbb {Z} \}$ .

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 $\mathbb {R} ^{n}$ 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.

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 $f : X \to Y$ .^[2]