Inverse of Real Square Function on Positive Reals

Theorem
Let $f: \R_{\ge 0} \to R_{\ge 0}$ be the restriction of the real square function to the positive real numbers $\R_{\ge 0}$.

The inverse of $f$ is $f^{-1}: \R_{\ge 0} \times R_{\ge 0}$ defined as:
 * $\forall x \in \R_{\ge 0}: \map {f^{-1} } x = \sqrt x$

where $\sqrt x$ is the positive square root of $x$.

Proof
From Restriction of Real Square Mapping to Positive Reals is Bijection, $f$ is a bijection.

By definition of the positive square root:
 * $y = \sqrt x \iff x = y^2$

for $x, y \in \R_{\ge 0}$.

Hence the result.