Domain of a function

A function f from X to Y. The set of points in the red oval X is the domain of f.The set of points in the blue oval Y is the codomain of f.The set of points in the yellow oval is the range of f.

In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by dom ⁡ ( f ) {\displaystyle \operatorname {dom} (f)} or dom ⁡ f {\displaystyle \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 : X → Y {\displaystyle 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 : X → Y {\displaystyle 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 {\displaystyle 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 : X → Y {\displaystyle f\colon X\to Y} to A {\displaystyle A}, where A ⊆ X {\displaystyle A\subseteq X}, is written as f | A : A → Y {\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 f {\displaystyle f} defined by f ( x ) = 1 x {\displaystyle f(x)={\frac {1}{x}}} cannot be evaluated at 0. Therefore, the natural domain of f {\displaystyle f} is the set of real numbers excluding 0, which can be denoted by R ∖ { 0 } {\displaystyle \mathbb {R} \setminus \{0\}} or { x ∈ R : x ≠ 0 } {\displaystyle \{x\in \mathbb {R} :x\neq 0\}}.
The piecewise function f {\displaystyle f} defined by f ( x ) = { 1 / x x ≠ 0 0 x = 0 , {\displaystyle f(x)={\begin{cases}1/x&x\not =0\\0&x=0\end{cases}},} has as its natural domain the set R {\displaystyle \mathbb {R} } of real numbers.
The square root function f ( x ) = x {\displaystyle f(x)={\sqrt {x}}} has as its natural domain the set of non-negative real numbers, which can be denoted by R ≥ 0 {\displaystyle \mathbb {R} _{\geq 0}}, the interval [ 0 , ∞ ) {\displaystyle [0,\infty )}, or { x ∈ R : x ≥ 0 } {\displaystyle \{x\in \mathbb {R} :x\geq 0\}}.
The tangent function, denoted tan {\displaystyle \tan }, has as its natural domain the set of all real numbers which are not of the form π 2 + k π {\displaystyle {\tfrac {\pi }{2}}+k\pi } for some integer k {\displaystyle k}, which can be written as R ∖ { π 2 + k π : k ∈ Z } {\displaystyle \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 R n {\displaystyle \mathbb {R} ^{n}} or the complex coordinate space C n . {\displaystyle \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 R n {\displaystyle \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.

Notes