Maximal domain for composite functions. 
Question $\mathbf 5$
If $f:(-\infty,1)\to R$, $f(x)=2\log_{\,e}(1-x)$ and $g:[-1,\infty)\to R$, $g(x)=3\sqrt{x+1}$, then the maximal domain of the function $f+g$ is
$\textbf{A.}\quad[-1,1)$ 
$\textbf{B.}\quad(1,\infty)$
$\textbf{C.}\quad(-1,1]$
$\textbf{D.}\quad(-\infty,-1]$
$\textbf{E.}\quad R$

It it correct to write the new domain as $[-1,1)$ because the values of $x$ are common to both domains? 

Yes. In general, if $f: A \to B$ and $g : C \to D$, then $f + g$ has domain $A\cap C$.
To see why this is the case, recall that the function $f + g$ is defined by $(f+g)(x) := f(x) + g(x)$. So $(f+g)(x)$ is defined only when both $f(x)$ and $g(x)$ are defined, so $x \in A$ and $x \in C$.
