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