Image of Subset under Mapping/Examples/Image of -3 to 2 under x^4-1
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Example of Image of Element under Mapping
Let $f: \R \to \R$ be the mapping defined as:
- $\forall x \in \R: \map f x = x^4 - 1$
The image of the closed interval $\closedint {-3} 2$ is:
- $f \closedint {-3} 2 = \closedint {-1} {80}$
Proof
Trivially, by differentiating $x^4 - 1$ with respect to $x$:
- $f' = 4 x^3$
and equating $f'$ to $0$, the minimum of $\Img f$ is seen to occur at $\map f 0 = -1$.
As $0 \in \closedint {-3} 2$, it can be seen that the minimum of $f \closedint {-3} 2$ is $-1$.
As $f$ is strictly increasing on $x > 0$ and strictly decreasing on $x < 0$, it suffices to inspect the images of the endpoints $-3$ and $2$.
Thus:
- $\map f {-3} = \paren {-3}^4 - 1 = 81 - 1 = 80$
- $\map f 2 = 2^4 - 1 = 16 - 1 = 15$
The result follows.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $4$: Mappings: Exercise $3 \ \text {(ii)}$