# 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