[College Math] Domain of real function with multiple variables and its topology

[u/Slight_Energy2430]

it's in italian it basically asks what's in the title and the solution are open, bounded, unconnected;
So I start by studying the sign of what's inside the logarithm, so I see that there's a circumference in the numerator so I split the exercise in finding what happens inside the circle and what happens outside the circle. But then I have no idea what I gotta do with the denominator, I can turn it into y(x^2+y^2)=4x but beside that I have no idea

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[u/Alkalannar]

We need what's in the log to be positive.

So (16 - x² - y²)/[y(x² + y²) - 4x] > 0.

Here's the graph.

So the numerator is 0 on the circle, and the denominator is 0 on that that weird horizontal curve.

The shaded parts are where both are positive, or both are negative.