Inverse Element/Examples/Square Root Function

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Examples of Inverse Elements

Let $\struct {\CC, \circ}$ be the monoid of all real functions $\CC$ under composition $\circ$ over the closed real interval $\closedint 0 1$.


Not all elements of $\CC$ have an inverse mapping, but in particular let $f$ be defined as:

$\forall x \in \closedint 0 1: \map f x = x^2$

$\map f x$ is in fact a bijection and has inverse mapping $\inv f x$ defined as:

$\forall x \in \closedint 0 1: \inv f x = \sqrt x$


Proof


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Sources

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