Domain of $h(x,y)=\frac{1-xy}{1+x^2y^2}+\ln \left(\frac{1-xy}{1+x^2y^2}\right)$

2018-03-12 20:24:39

I am asked the following question:

Let $g$ be the function given by $g(t)=t+\ln t$

and $f$ the two variable function given by $\displaystyle f(x,y)=\frac{1-xy}{1+x^2y^2}$. What is the expression of $h(x,y)=g(f(x,y))$? Where is $h$ continuous?

This question has no solution on the textbook, so I would like to check my answer on this forum.

Answer

By evaluating the composite function I get

$$h(x,y)=\frac{1-xy}{1+x^2y^2}+\ln \left(\frac{1-xy}{1+x^2y^2}\right)$$

In order to stipulate it's domain, we should have

\begin{align*}

1+x^2y^2 &\neq 0 \quad \text{(always)}\\

\\

\frac{1-xy}{1+x^2y^2} &> 0

\end{align*}

So what we need to consider is every $x$ and $y$ where $xy < 1$

Is that correct?