Domain of A Function - Formal Definition

Formal Definition

Given a function f:X→Y, the set X is the domain of f; the set Y is the codomain of f. In the expression f(x), x is the argument and f(x) is the value. One can think of an argument as an input to the function, and the value as the output.

The image (sometimes called the range) of f is the set of all values assumed by f for all possible x; this is the set . The image of f can be the same set as the codomain or it can be a proper subset of it. It is in general smaller than the codomain; it is the whole codomain if and only if f is a surjective function.

A well-defined function must carry every element of its 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 of all real numbers, cannot be its domain. In cases like this, the function is either defined on or the "gap is plugged" by explicitly 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 its domain is .

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.