Example of Bijective Restrictions
Let $f: \R \to \R$ be the real function defined as:
- $\forall x \in \R: \map f x = x^2 - 4 x + 5$
|\(\ds f_1: \hointl \gets 2\)||\(\to\)||\(\ds \hointr 1 \to\)|
|\(\ds f_2: \hointr 2 \to\)||\(\to\)||\(\ds \hointr 1 \to\)|
- $\Img f = \hointr 1 \to$
Thus a surjective restriction of $f$ can be found as:
- $g: \R \to \hointr 1 \to: \map g x = x^2 - 4 x + 5$
It remains to show that $f_1$ and $f_2$ are injective.
As this is the only stationary point of $f$, it follows that:
- $\map f x$ is strictly decreasing on $\hointl \gets 2$
- $\map f x$ is strictly increasing on $\hointl 2 \to$
From Strictly Monotone Mapping with Totally Ordered Domain is Injective it follows that both $f_1$ and $f_2$ are injections.
Hence, by definition, $f_1$ and $f_2$ are bijective restrictions of $f$.