Why is the absolute sign needed in the definition of a bounded function A function $f$ is bounded if there exists a real number $M$ such that $|f(x)| \le M$ for all $x \in \operatorname{dom}(f)$. Why is the absolute sign needed?

I think the definition should be
\begin{equation}
|f(x)| \leq M.
\end{equation}
We need the absolute value so that $f$ is also bounded from below. For example, if we only had 
\begin{equation}
f(x) \leq M,
\end{equation}
$f$ can be unbounded below. That is, we can have $f$ approaching $-\infty$. 
