# Domain (mathematics)

(Redirected from Function domain)

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

X, the set of input values, is called the domain of f, and Y, the set of possible output values, is called the codomain. The range of f is the set of all actual outputs {f(x) : x in the domain}. Beware that sometimes the codomain is incorrectly called the range because of a failure to distinguish between possible and actual values.

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 SA, is written g |S : S → B.

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