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:


 * $\forall x \in \R: f \paren x = x^4 - 1$

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 $x^4 - 1$ $x$:
 * $f' = 4 x^3$

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.