Finding the domain of a three variable function

Question

I have a problem trying to solve the next excersice:

find the domain of the following function,

$f(x,y,z)=\frac{\sqrt{x^2z-3xz^2+3z^3}}{xy-x^2-y^2-1}$

I know that the domain should be:

$Domain=\{(x,y,z)|x^2z-3xz^2+3z^3 \geq 0\} \setminus \{(x,y,z)|xy-x^2-y^2-1 = 0 \} $

The thing is I have some problems finding the x,y for which:

$xy-x^2-y^2-1 = 0$ , It seems like $xy-x^2-y^2-1\neq0$ for all $x,y \in R$, but again I have some problems given the right argument.

I know it might seems a bit easy but I've been struggling for some time with this.

I would really apreciate any advice or hint you can give me.

Answer 1

Note that we can write

$$\begin{align} xy-x^2-y^2-1&=-(x-y/2)^2-3y^2/4-1\\\\ &\le -1\end{align}$$

for all $x$ and $y$.

Hence the denominator of $f(x,y,z)$ can never be zero.