# Definition:Extension of Mapping

## Definition

As a mapping is, by definition, also a relation, the definition of an extension of a mapping is the same as that for an extension of a relation:

Let:

$f_1 \subseteq X \times Y$ be a mapping on $X \times Y$
$f_2 \subseteq S \times T$ be a mapping on $S \times T$
$X \subseteq S$
$Y \subseteq T$
$f_2 \restriction_{X \times Y}$ be the restriction of $f_2$ to $X \times Y$.

Let $f_2 \restriction_{X \times Y} = f_1$.

Then $f_2$ extends or is an extension of $f_1$.

## Examples

### Extension of Square Function on Natural Numbers

Let $f: \N \to \N$ be the mapping defined as:

$\forall n \in \N: \map f n = n^2$

Let $h: \R \to \R$ be the mapping defined as:

$\forall x \in \R: \map h x = x^2$

Then $h$ is a extension of $f$.