Step 1: Understand the sum rule.
For $(f+g)(x)=f(x)+g(x)$ to make sense, both $f$ and $g$ must be defined at that same $x$, so we intersect their domains.
Step 2: Domain of $f(x)=\sqrt{2-x^2}$.
A square root needs a non-negative inside, so $2-x^2\ge 0$, that is $x^2\le 2$.
Step 3: Solve the inequality for $f$.
This gives $-\sqrt{2}\le x\le \sqrt{2}$, so the domain of $f$ is $[-\sqrt2,\ \sqrt2]$.
Step 4: Domain of $g(x)=\log(1-x)$.
A logarithm needs a strictly positive argument, so $1-x>0$, giving $x<1$, that is $(-\infty,1)$.
Step 5: Intersect the two domains.
We need $-\sqrt2\le x\le \sqrt2$ and $x<1$ together. Since $1<\sqrt2$, the binding upper limit is $x<1$, while the lower limit stays at $-\sqrt2$.
Step 6: State the common region.
The overlap is all $x$ with $-\sqrt2\le x<1$, written $[-\sqrt2,\ 1)$.
\[ \boxed{[-\sqrt{2},\ 1)} \]