I have recently begun to see examples of people and literature saying statements such as $f(x) \in \Theta(...)$ but what does it mean for one function to be inside or a member of another?
(i.e. I understand that's probably not what it means in this case, but what is this alternative meaning used for functions?)
Unfortunately, there are massive inconsistencies with how big O (and related) notation is used, leading to some confusion.
In this case $\Theta(g(x))$ is the set of functions that are asymptotically bounded below and above by $g$.
Therefore, if $f\in \Theta(g)$ that means that $f$ is in that set, i.e. it is bounded below and above by $g$.
A more common notation is $f = \Theta(g(x))$ (see wikipedia), but as the latter is a set of functions, a more set-theoretical notation is to write $f \in \Theta(g(x))$ instead. It says that $f$ belongs to a certain set of functions vis-à-vis $g$.
Just to formalise the other answers slightly, we say that
$f(x) \in \Theta(g(x))$
if and only if
$\exists \: c_1, c_2, N \in \mathbb{R}_{>0}$
such that
$0 < c_1 g(x) \leq f(x) \leq c_2 g(x) \quad \forall x \geq N$.
That $\Theta$ is an example of a construct in the area loosely described as "big-O analysis" concerned with how fast functions grow.
The statement $$ f(x) \in \Theta(g(x)), $$ sometimes written $$ f(x) = \Theta(g(x)), $$ means $f$ and $g$ grow at the same rate.
https://cathyatseneca.gitbooks.io/data-structures-and-algorithms/analysis/notations.html
