I'm not sure what this is exactly asking Without using words of negation, write the meaning of : "f is not an increasing function"
I did:
$$"f\ is\ not\ an\ increasing\ function" \ \equiv\ "f\ is\ a\ decreasing\ function"$$
Is this what it's asking or am I completely missing it?

No this is not true, for example any function from $\mathbb{R}$ to itself which increases on one interval and decreases on another is neither increasing nor decreasing.
It should be noted that "nonincreasing" is actually a perfectly valid adjective for describing a function, but that is probably not what it being sought after.
One way to write this, would to be to use the definition of increasing functions: a function $f$ is said to be increasing if $x>y\implies f(x)\geq f(y)$, or in other words $\forall x,y$ such that $x>y$, it follows that $f(x)\geq f(y)$.
The negation of this statement is therefore, $\exists x,y$ such that $x>y$, and $f(x)<f(y)$.
