Domain of a Function

The domain of a function or relation is the set A in the mapping $$ f:A \rightarrow B $$

The domain is the set from which the function takes its inputs.

The set B is the target set and is called the codomain of the function.

The difference between domain and domain of definition

The elements of A (the domain) that are actually mapped to at least one element of B (the codomain) form a subset known as the domain of definition (or existence set) of the function f.

In general, the domain does not coincide with the domain of definition:

$$ f: D(f) \subseteq A \rightarrow B $$

The domain is the ambient space, while the domain of definition is the portion where the function is effectively defined.

Note. In some texts, the term “domain” is used differently: it is taken to mean precisely the set of elements of A that are mapped to at least one element of B under the relation R. In that usage, the domain coincides with the domain of definition.

A Practical Example

Consider two finite sets X and Y:

$$ X = \{1,2,3,4,5 \} $$

$$ Y = \{1,2,3,4,5,6,7,8,9 \} $$

and the relation/function:

$$ f: y = x^2 $$

The domain of this relation is the entire set X:

$$ \text{Dom}(f) = \{1,2,3,4,5 \} $$

The domain of definition is a subset of the domain:

$$ D_f = \{1,2,3 \} \subseteq X $$

These are precisely the elements of X whose squares lie in the codomain Y.

$$ f: 1^2 \mapsto 1 $$

$$ f: 2^2 \mapsto 4 $$

$$ f: 3^2 \mapsto 9 $$

Example 2

Consider the real function:

$$ f: y = \sin(x) $$

Here both the domain and the codomain are the set of real numbers:

$$ f: \mathbb{R} \rightarrow \mathbb{R} $$

In this case, the domain of definition coincides with the domain, since the sine function is defined for all real numbers.