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

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## 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 $f$:

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

\(\ds f_2: \hointr 2 \to\) | \(\to\) | \(\ds \hointr 1 \to\) |

## Proof

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 $f$.

$\blacksquare$

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): Chapter $4$: Mappings: Exercise $12 \ \text{(iii)}$