Domain (mathematics)

de:Definitionsmenge fr:Ensemble de définition sv:definitionsmängd In mathematics, the domain of a function is the set of all input values to the function.

Given a function <math>f\colon A\rightarrow B<math>, 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, or f(A).

A well-defined function must map every element of the domain to an element of its codomain. So, for example, the function:

<math>f\colon\,x\mapsto 1/x<math>

has no valid value for f(0). It is thus not a function on the set R of real numbers; R can't be its domain. It is usually either defined as a function on R \ {0}, or the "gap" is plugged by specifically defining f(0); for example:

<math>f\colon x\mapsto 1/x,\quad x\neq0<math>

<math>f\colon0\mapsto 0<math>

The domain of given function can be restricted to a subset. Suppose that <math>g\colon A\to B<math>, and <math>S\subset A<math>. Then the restriction of g to S is written:

<math>g|_S\colon S\to B<math>

See also