Right Inverse Mapping/Examples
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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.