Bijective Restriction/Examples/x^2-4x+5

From ProofWiki
Jump to navigation Jump to search

Example of Bijective Restrictions

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = x^2 - 4 x + 5$

The following real functions are bijective restrictions of $f$:

\(\ds f_1: \hointl \gets 2\) \(\to\) \(\ds \hointr 1 \to\)
\(\ds f_2: \hointr 2 \to\) \(\to\) \(\ds \hointr 1 \to\)


From Image of $\map f x = x^2 - 4 x + 5$, the image of $f$ is given by:

$\Img f = \hointr 1 \to$

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.

It is established in Image of $\map f x = x^2 - 4 x + 5$ that $f$ has a minimum at $x = 2$.

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 $f$.