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?