Describing multivariable functions So I am presented with the following question:
Describe and sketch the largest region in the $xy$-plane that corresponds to the domain of the function:
$$g(x,y) = \sqrt{4 - x^2 - y^2} \ln(x-y).$$
Now to be I can find different restrictions like $4 - x^2 - y^2 \geq 0$... but I'm honestly not even sure where to begin this question! Any help?

When is $g(x,y)$ defined and takes a real value?  Well, the square root has to be of a positive number, so $x^2+y^2\le4$.  Also, when you are taking $\log(x-y)$, for the logarithm to be defined, $x-y$ must be greater than $0$.  Thus, the domain of $g(x,y)$ consists of all points $(x,y)$ such that
(1) $x^2+y^2\le4$ and
(2) $y<x$
Sketch (1) and (2) on an $xy$ plane.  (1) is a filled-in circle with radius $2$ centered around the origin (all points on the boundary are points in the domain).  (2) is all points strictly below the line $y=x$ (all points on the boundary are not points in the domain).  For $g(x,y)$ to be defined, both (1) and (2) must be satisfied, so we must take the intersection of these two figures.
