Domain of a function

In layman's terms, the domain of a function can generally be thought of as "what x can be".

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.

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.

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.

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

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

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).