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 $$\operatorname{dom}(f)$$ or $$\operatorname{dom }f$$, 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 $$f\colon X\to Y$$, 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 $$f\colon X\to Y$$, 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 $$f \colon X \to Y$$ to $$A$$, where $$A\subseteq X$$, is written as $$\left. f \right|_A \colon A \to Y$$.

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 $X$. In many contexts, a partial function is called simply a function, and its natural domain is called simply its domain.

Examples
1/x&x\not=0\\ 0&x=0 \end{cases},$$ has as its natural domain the set $$\mathbb{R}$$ of real numbers.
 * The function $$f$$ defined by $$f(x)=\frac{1}{x}$$ cannot be evaluated at 0. Therefore, the natural domain of $$f$$ is the set of real numbers excluding 0, which can be denoted by $$\mathbb{R} \setminus \{ 0 \}$$ or $$\{x\in\mathbb R:x\ne 0\}$$.
 * The piecewise function $$f$$ defined by $$f(x) = \begin{cases}
 * 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 $$\mathbb R_{\geq 0}$$, the interval $$[0,\infty)$$, or $$\{x\in\mathbb R:x\geq 0\}$$.
 * The tangent function, denoted $$\tan$$, has as its natural domain the set of all real numbers which are not of the form $$\tfrac{\pi}{2} + k \pi$$ for some integer $$k$$, which can be written as $$\mathbb R \setminus \{\tfrac{\pi}{2}+k\pi: k\in\mathbb Z\}$$.

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 $$\R^n$$ 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 $$\R^{n}$$ 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 $Y$, 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 $f: X → Y$.