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 ${\displaystyle \operatorname {dom} (f)}$ or ${\displaystyle \operatorname {dom} f}$, where f is the function. In layman's terms, the domain of a function can generally be thought of as "what x can be".

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, f(x) = √x, whose domain consists of all nonnegative real numbers

More precisely, given a function ${\displaystyle f\colon X\to Y}$, 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 ${\displaystyle f\colon X\to Y}$, 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 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 ${\displaystyle f\colon X\to Y}$ 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 ${\displaystyle f}$ is the set of real numbers excluding 0, which can be denoted by ${\displaystyle \mathbb {R} \setminus \{0\}}$ or ${\displaystyle \{x\in \mathbb {R} :x\neq 0\}}$.
• The piecewise function ${\displaystyle f}$ defined by ${\displaystyle f(x)={\begin{cases}1/x&x\not =0\\0&x=0\end{cases}},}$ has as its natural domain the set ${\displaystyle \mathbb {R} }$ of real numbers.
• The square root function ${\displaystyle f(x)={\sqrt {x}}}$ has as its natural domain the set of non-negative real numbers, which can be denoted by ${\displaystyle \mathbb {R} _{\geq 0}}$, the interval ${\displaystyle [0,\infty )}$, or ${\displaystyle \{x\in \mathbb {R} :x\geq 0\}}$.
• The tangent function, denoted ${\displaystyle \tan }$, has as its natural domain the set of all real numbers which are not of the form ${\displaystyle {\tfrac {\pi }{2}}+k\pi }$ for some integer ${\displaystyle k}$, which can be written as ${\displaystyle \mathbb {R} \setminus \{{\tfrac {\pi }{2}}+k\pi :k\in \mathbb {Z} \}}$.

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 ${\displaystyle \mathbb {R} ^{n}}$ or the complex coordinate space ${\displaystyle \mathbb {C} ^{n}.}$

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

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)

Notes

1. "Domain, Range, Inverse of Functions". Easy Sevens Education. 10 April 2023. Retrieved 2023-04-13.
2. Eccles 1997, p. 91 (quote 1, quote 2); Mac Lane 1998, p. 8; Mac Lane, in Scott & Jech 1971, p. 232; Sharma 2010, p. 91; Stewart & Tall 1977, p. 89

References

• Bourbaki, Nicolas (1970). Théorie des ensembles. Éléments de mathématique. Springer. ISBN 9783540340348.
• Eccles, Peter J. (11 December 1997). An Introduction to Mathematical Reasoning: Numbers, Sets and Functions. Cambridge University Press. ISBN 978-0-521-59718-0.
• Mac Lane, Saunders (25 September 1998). Categories for the Working Mathematician. Springer Science & Business Media. ISBN 978-0-387-98403-2.
• Scott, Dana S.; Jech, Thomas J. (31 December 1971). Axiomatic Set Theory, Part 1. American Mathematical Soc. ISBN 978-0-8218-0245-8.
• Sharma, A. K. (2010). Introduction To Set Theory. Discovery Publishing House. ISBN 978-81-7141-877-0.
• Stewart, Ian; Tall, David (1977). The Foundations of Mathematics. Oxford University Press. ISBN 978-0-19-853165-4.