Right Inverse Mapping/Examples/Real Square Function to Non-Negative Reals

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Example of Right Inverse Mapping

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:

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


Sources