In mathematics, the domain of a function is the set of all input values to the function.

Given a function f : A → B, the set A is called the domain, or domain of definition of f.

The set of all values in the codomain that f maps to is called the range of f, written f(A).

A well-defined function must map every element of the domain to an element of its codomain. For example, the function f defined by

f(x) = 1/x

has no value for f(0). Thus, the set R of real numbers cannot be its domain. In cases like this, the function is usually either defined on R \ {0}, or the "gap" is plugged by specifically defining f(0). If we extend the definition of f to

f(x) = 1/x, for x ≠ 0

f(0) = 0,

then f is defined for all real numbers and we can choose its domain to be R.

Any function can be restricted to a subset of its domain. The restriction of g : A → B to S, where S ⊆ A, is written g |_S : S → B.

See also

• codomain
• range (mathematics)
• injective function
• surjective function
• bijective function