Domain, range and zeros of $f(x,y)=\frac{\sqrt{4x-x^2-y^2}}{x^2y^2-4xy^2+3y^2}$ Given the following function with two variables:
\begin{equation}
    \frac{\sqrt{4x-x^2-y^2}}{x^2y^2-4xy^2+3y^2}
\end{equation}
I need to find a) the domain for the above function.
Can anyone give me a hint on how to find the domain in f?
I already know that:
\begin{equation}
    dom f = \{ f(x,y) ∈ \mathbb{R}^2\backslash(x^2y^2-4xy^2+3y^2=0) : \sqrt{4x-x^2-y^2} \geq 0 \}
\end{equation}
But of course this needs to be written in a more simpler form.
During class we solve simpler functions  like without fractions and roots, so I don't have anything that can help me.
After this I also need to find:


*

*b) zeros of the function  

*c) Calculating Algebraically, the range of the function:
\begin{equation}
T(x,y) = \sqrt{4x-x^2-y^2}
\end{equation}

*d) Extreme values of the function T


I'm of course not expecting the complete solution but something like a kick start.


Can anyone give me a hint on how to find the domain in f?

You need to find all the values for which $x^2 y^2-4xy^2+3y^2$ is not equal to zero and $4x-x^2-y^2$ is positive or zero.

zeros of the function

Well you just need to find where $\sqrt{4x-x^2-y^2}=0$, which is just where $4x-x^2-y^2=0$.
