Function Given by Equation I

Petra was supposed to find out if the functions $f$, $g$, and $h$ are equal. $$ f(x)=\frac{x^2-1}{x+1};~~~g(x)=x-1;~~~h(x)=2^{\log_2⁡(x-1) } $$ She simplified the equations of the functions $f$ and $h$ as follows: $$ \frac{x^2-1}{x+1}=\frac{(x-1)(x+1)}{x+1}=x-1 $$ $$ 2^{\log_2⁡(x-1)} =x-1 $$ Then she stated that all the given functions are the same (i.e., they equal): $$ f=g=h $$

Her classmates commented on her solution:

Joey argues that to determine the equality of functions, it is not enough to show that their expressions are equal. We need to know their domains as well, which are not given. Therefore, the problem is unsolvable.

Erica says that domains are not needed. If the functions have the same equations after simplification, then they are equal. She says that Petra solved the problem correctly.

Sarah says that if the domains are not given, we have to find them. She determined the domains as: $$D(f)=D(g)=D(h)=\mathbb{R} -\{ -1 \}$$ She is convinced that the functions are equal.

Charles thinks that equations of functions should not be simplified. But still, he agrees that $f=g=h$. If, for example, we substitute $x=5$, we can see that $f(5)=g(5)=h(5)=4$.

Who of them is right?