I have the function
$f(x,y)=e^{2x}+\sin(x+\ln(y+1))-1$
and I need to prove the existence of a function $g(x): \; ]-e,e[ \; \to \mathbb{R}$ with $e>0$ so $f(x,g(x))=0$ for all $x \in\;]-e,e[$ and calculate $g'(0)$.
I use the implicit function theorem to prove the existence of $y=g(x)$ around $(0,0)$ but I have doubts about being a function mapping $]-e,e[\; \to \mathbb{R}$. I found that the domain of $y$ is $]-1, \infty[$ because of the logarithm so wouldn't $g(x)$ be a function mapping $]-e,e[ \; \to \; ]-1, \infty[$ ?