Maximal domain for composite functions.

Question

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?

Answer 1

Yes, you are completely correct.

In addition to what you said, any value which does not satisfy the domain is not defined in either $f$ or $g$.

Answer 2

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 $x \in C$.