Natural Domain

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.

More precisely, given a function $f\colon X\to Y$, the domain of is . Note that 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 subsets of $\R$, 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, and the set of values attained by the function (which is a subset of ) is called its range or image.

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 word "domain" is used with other related meanings in some areas of mathematics. In topology, a domain is a connected open set. In real and complex analysis, a domain is an open connected subset of a real or complex vector space. In the study of partial differential equations, a domain is the open connected subset of the Euclidean space $\mathbb^$ where a problem is posed (i.e., where the unknown function(s) are defined).

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

* Attribute domain
* Bijection, injection and surjection
* Codomain
* Domain decomposition
* Effective domain
* Image (mathematics)
* Lipschitz domain
* Naive set theory
* Support (mathematics)

Notes

References

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{{Mathematical logic

Functions and mappings
Basic concepts in set theory