Answer
$
g(x)=5\cdot\left(2^{-1/2}\right)^x
$
Work Step by Step
We are given $g(0)=5$ and $g(-2)=10$, so $a=5$. Then $g(x)=a b^x=5 b^x$, find $b$ from
$$
\begin{aligned}
5 b^{-2} & =g(-2)= 10 \\
b^{-2} & =2 \\
b & =2^{-1/2}.
\end{aligned}
$$ Hence $$
g(x)=5\cdot\left(2^{-1/2}\right)^x
$$
