Domain of a function

File:Codomain2.SVG

A function Template:Mvar from Template:Mvar to Template:Mvar. The set of points in the red oval Template:Mvar is the domain of Template:Mvar.

File:Arcsine Arccosine.svg

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

In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by $\mathrm{dom}(f)$ or $\mathrm{dom}f$, where Template:Math 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 $f:X\to Y$, the domain of Template:Math is Template:Math. 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 Template:Math and Template:Math are both sets of real numbers, the function Template:Math can be graphed in the Cartesian coordinate system. In this case, the domain is represented on the Template:Math-axis of the graph, as the projection of the graph of the function onto the Template:Math-axis.

For a function $f:X\to Y$, the set Template:Math is called the codomain: the set to which all outputs must belong. The set of specific outputs the function assigns to elements of Template:Math is called its range or image. The image of $f$ is a subset of Template:Math, shown as the yellow oval in the accompanying diagram.

Any function can be restricted to a subset of its domain. The restriction of $f:X\to Y$ to $A$, where $A\subseteq X$, is written as ${f|}_A:A\to Y$.

Natural domain

If a real function Template:Mvar 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 Template:Mvar. 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 $\{x\in \mathbb{ℝ}:x\ne 0\}$.
• The piecewise function $f$ defined by $f(x)=\{\begin{array}{ll}1/x& x=0\\ 0& x=0\end{array},$ 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 ${\mathbb{ℝ}}_{\ge 0}$, the interval $[0,\mathrm{\infty })$, or $\{x\in \mathbb{ℝ}:x\ge 0\}$.
• The tangent function, denoted $\tan$, has as its natural domain the set of all real numbers which are not of the form $\frac{\pi }{2}+k\pi$ for some integer $k$, which can be written as $\mathbb{ℝ}\setminus \{\frac{\pi }{2}+k\pi :k\in \mathbb{\mathbb{Z}}\}$.

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 Template:Mvar, in which case there is formally no such thing as a triple Template:Math. With such a definition, functions do not have a domain, although some authors still use it informally after introducing a function in the form Template:Math.

See also

• Argument of a function
• Attribute domain
• Bijection, injection and surjection
• Codomain
• Domain decomposition
• Effective domain
• Endofunction
• Image (mathematics)
• Lipschitz domain
• Naive set theory
• Range of a function
• Support (mathematics)