Inverse of Real Square Function on Positive Reals
Jump to navigation
Jump to search
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.
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $3$. Mappings: Exercise $3$