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

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Example of Image of Element under Mapping

Let $f: \R \to \R$ be the mapping defined as:

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


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

$\Img f = \hointr 1 \to$

and so $f$ is not a surjection.


Graphical Representation of $\map f x = x^2 - 4 x + 5$

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

Proof

By differentiating $x^2 - 4 x + 5$ twice with respect to $x$:

$f' = 2 x - 4$
$f' = 2 x - 4$
\(\ds f'\) \(=\) \(\ds 2 x - 4\)
\(\ds f''\) \(=\) \(\ds 2\)

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:

$\Img f = \hointr 1 \to$

$\blacksquare$


Also see


Sources