wikiwand.com

Domain of a function - Wikiwand

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".[1]

Thumb
A function f from X to Y. The set of points in the red oval X is the domain of f.
Thumb
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 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.

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.[2]

Loading related searches...