Image of Mapping/Examples/Image of x^2-4x+5

Example of Image of Element under Mapping
Let $f: \R \to \R$ be the mapping defined as:


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

The image of $f$ is the unbounded closed interval:


 * $\Img f = \hointr 1 \to$

and so $f$ is not a surjection.

Proof
By differentiating $x^2 - 4 x + 5$ twice $x$:
 * $f' = 2 x - 4$


 * $f' = 2 x - 4$

Equating $f'$ to $0$, a stationary point is found at $x = 2$.

Inspecting the sign of $f''$, it is noted that $f'$ is increasing everywhere.

Hence the stationary point at $x = 2$ is a minimum of $\Img f$.

This is the only stationary point, so it can be stated that the minimum of $f$ occurs at $x = 2$.

We have that:
 * $f \paren 2 = 2^2 - 4 \times 2 + 5 = 4 - 8 + 5 = 1$

As $f$ is strictly increasing on $x > 2$ and strictly decreasing on $x < 2$, it is seen that $f$ is unbounded above.

Thus:
 * $\Img f = \hointr 1 \to$

Also see

 * Bijective Restrictions of $f \paren x = x^2 - 4 x + 5$