Domain (function)

In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by $\operatorname(f)$ or $\operatornamef$, where 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 is . 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 and are both sets of real numbers, the function can be graphed in the Cartesian coordinate system. In this case, the domain is represented on the -axis of the graph, as the projection of the graph of the function onto the -axis.

For a function $f\colon X\to Y$, the set is called the ''codomain'': the set to which all outputs must belong. The set of specific outputs the function assigns to elements of is called its '' range'' or ''image''. The image of f is a subset of , 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 $\left. f \_A \colon A \to Y$.

Natural domain

If a real function 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 . 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 $f(x)=\frac$ 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 \setminus \$ or $\$.
* The piecewise function $f$ defined by $f(x) = \begin 1/x&x\not=0\\ 0&x=0 \end,$ has as its natural domain the set $\mathbb$ 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 $\mathbb R_$, the interval ,\infty), or $\$.
* The tangent function, denoted $\tan$, has as its natural domain the set of all real numbers which are not of the form $\tfrac + k \pi$ for some integer $k$, which can be written as $\mathbb R \setminus \$.

Other uses

The term ''domain'' is also commonly used in a different sense in mathematical analysis: a '' Domain (mathematical analysis), 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 $\R^n$ or the complex coordinate space $\C^n.$

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 $\R^$ 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 , in which case there is formally no such thing as a triple . With such a definition, functions do not have a domain, although some authors still use it informally after introducing a function in the form ., p. 91 ( quote 1 quote 2; , p. 8 Mac Lane, in , p. 232 , p. 91 , p. 89/ref>

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

References

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{{Mathematical logic

Functions and mappings
Basic concepts in set theory