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]

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 ${\displaystyle \left.f\right|_{A}\colon A\to Y}$.