Image of Subset under Mapping/Examples/Image of 1 to 2 under x^2-x-2
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Example of Image of Subset under Mapping
Let $f: \R \to \R$ be the mapping defined as:
- $\forall x \in \R: \map f x = x^2 - x - 2$
The image of the closed interval $\closedint {-3} 2$ is:
- $f \closedint 1 2 = \closedint {-2} 0$
Proof
Trivially, by differentiating $x^2 - x - 2$ with respect to $x$:
- $f' = 2 x - 1$
It is seen that, on $\closedint 1 2$, $f$ is strictly increasing.
Hence it suffices to inspect the images of the endpoints $1$ and $2$.
Thus:
- $\map f 1 = 1^2 - 1 - 1 = -2$
- $\map f 2 = 2^2 - 2 - 2 = 0$
The result follows.
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 6$: Functions