Image of Mapping/Examples/Image of x^4-1

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 $f$ is the unbounded closed interval:

$\Img f = \hointr {-1} \to$

and so $f$ is not a surjection.

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 $f \paren 0 = -1$.

Thus it can be seen that the minimum of $\Img f$ is $-1$.

As $f$ is strictly increasing on $x > 0$ and strictly decreasing on $x < 0$, it is seen that $f$ is unbounded above.

Thus:

$\Img f = \hointr {-1} \to$

As $\Img f \subset \R$ and $\Img f \ne \R$, it follows by definition that $f$ is not surjective.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $4$: Mappings: Exercise $3 \ \text {(iii)}$