Find the Range and Domain of the following function The function is:
$f(x,y) = \frac{2}{\sqrt{3-x}} + \frac{1}{\sqrt{4-y}}$
I have found the domain and the Range intuitively. 
But how would I formally prove that my assumption of the Range and Domain is true?
 A: You can input Your function in WolframAlpha
From alternate form the domain and range is quite clear.
A: Informally, to find the domain, we simply examine sub-parts of the expression, looking for potentially illegal operations.  When we see a sub-part like $\sqrt{4-y}$, for example, we know that $4-y$ better be positive.  We find all such restrictions and then form their conjunction.
As far as agnis's WolframAlpha suggestion goes, it's hard to imagine a more "formal" computation than what the computer might do.  Your function
2*(3 - x)^(-1/2) + (4 - y)^(-1/2)

can be represented as a graph theoretical tree:

We now traverse the tree.  Some nodes lead to restrictions; others do not.  For example, a Plus node is no difficulty, since we can always add any two real numbers.  We just record the visit and move on.  A Power node can lead to a problem.  Specifically, Power[expression,exponent] leads to the restriction expression!=0, whenever exponent<0.  Furthermore, expression must be non-negative if exponent is a rational number.  Thus, for example, the Power node on the right leads to the inequality $4-y>0$.  Finally, we take the intersection of all these parts.
A: I suppose the function is defined on $ \mathbb{R}^2 $; thus the domain has to contain elements of $ \mathbb{R}^2 $ such that the function gives a result in $ \mathbb{R} $.
For the range, you can observe that for every element in the domain of $f$ you have $f(x,y)>0$, then you notice that as $ x \rightarrow 3 $, $ y \rightarrow 4 $, $ f(x,y) $ takes arbitrary large values. Since the function is continuous on its domain, the range is thus $ (0,+\infty)$. 
