Right Inverse Mapping/Examples

Examples of Right Inverse Mappings

Real Square Function to $\R_{\ge 0}$

Let $f: \R \to \R_{\ge 0}$ be the real square function whose codomain is the set of non-negative reals:

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

From Real Square Function to $\R_{\ge 0}$, $f$ is a surjection.

Hence it has a right inverse $g: \R_{\ge 0} \to \R$ which, for example, can be defined as:

$\forall x \in \R_{\ge 0}: \map g x = +\sqrt x$

This right inverse is not unique.

For example, the mapping $h: \R_{\ge 0} \to \R$ defined as:

$\forall x \in \R_{\ge 0}: \map h x = -\sqrt x$

is also a right inverse, as is the arbitrarily defined mapping $j: \R_{\ge 0} \to \R$ defined as:

$\forall x \in \R_{\ge 0}: \map j x = \begin {cases} \sqrt x & : x \le 5 \\ -\sqrt x & : x > 5 \end {cases}$

Real Part of Complex Number

Let $f: \C \to \R$ be the mapping:

$\forall z \in \C: \map f z = \map \Re z$

From Real Part as Mapping is Surjection, $f$ is a surjection.

Hence it has a right inverse $g: \R \to \C$ which, for example, can be defined as:

$\forall x \in \R: \map g x = x + i$

This right inverse is not unique.

For example, the mapping $h: \R \to \C$ defined as:

$\forall x \in \R: \map h x = x - i$

is also a right inverse.