Mapping/Examples/x^3-x on Real Numbers

From ProofWiki
Jump to navigation Jump to search

Example of Mapping

Let $f: \R \to \R$ be the mapping defined on the set of real numbers as:

$\forall x \in \R: \map f x = x^3 - x$

Then $f$ is a surjection but not an injection.


Let $y \in \R$.

As $x \to \infty$, we have that $y \to \infty$.

Similarly, as $x \to -\infty$, we have that $y \to -\infty$.

From Real Polynomial Function is Continuous, $f$ is continuous on $\R$.

It follows from the Intermediate Value Theorem that:

$\forall y \in \R: \exists x \in \R: y = \map f x$

Thus, by definition, $f$ is a surjection.


We have that:

\(\ds \map f 0\) \(=\) \(\ds 0^3 - 0\)
\(\ds \) \(=\) \(\ds 0\)
\(\ds \map f 1\) \(=\) \(\ds 1^3 - 1\)
\(\ds \) \(=\) \(\ds 0\)

demonstrating that $f$ is not an injection.