# Domain of a function

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

## 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 → Y.