Why do we use the term "such that" instead of just using "and"? Instead of saying "The function $f(x)=2x$ such that $x>0$"
why not just say "$f(x)=2x$ AND $x>0$"
logically it gives the exact same restriction.. so what is the benefit of adding the term "such that" over using logical AND?
 A: I would not say ‘The function $f(x)=2x$ such that $x>0$’; it’s very poor usage. So is ‘The function $f(x)=2x$ and $x>0$’. What is meant here is ‘The function $f$ on the set of positive real numbers defined by $f(x)=2x$’. There are shorter, more informal ways to say this, but they don’t include either of your formulations. One could, for example, say ‘The function $f(x)=2x$ restricted to the positive reals’, or even, a bit more sloppily, ‘The function $f(x)=2x$ on $x>0$’.
More generally, when such that is used correctly, it can almost never be replaced by and. The expression the object $X$ such that $P(x)$ means the object $X$ with the property $P$; $P$ is a property of $X$, part of the description of $X$, not some independent statement that can be attached with the logical conjunction and.
A: For this example I would say "The function $f(x)=2x$ for $x>0$".
Here, "for" is shorthand for "defined for all $x>0$".
A: Your correct that "such that" isn't appropriate, nor is "and": What we have, essentially, is a conditional:
IF $x\gt 0$, then $f(x) = 2x$
More formally, this can be written:
$$\forall x\Big((x\gt 0) \rightarrow (f(x) = 2x)\Big)$$
Alternatively, if you are working with a specific function and wanting to claim the existence of an $x\gt 0$ for which $f(x) = 2x$, you can write: 
There exists an x such that $(x\gt 0$ and $f(x)= 2x)$. 
That can be written: $$\exists x( x\gt 0 \land f(x) = 2x)$$
