Answer
$
g(x)=2\cdot\left(4\right)^x
$
Work Step by Step
We want to find $f(x)=ab^x$,given $g(1 / 2)=4$ and $g(1 / 4)=2 \sqrt{2}$. We now find $b$ and $a$.
$$
\begin{aligned}
\frac{a b^{1 / 2}}{a b^{1 / 4}} & =\frac{g(1/2)}{g(1/4)} \\
\frac{a b^{1 / 2}}{a b^{1 / 4}} & =\frac{2^2}{2^{3 / 2}} \\
b^{1 / 4} & =2^{1 / 2} \\
\left(b^{1 / 4}\right)^4 & =\left(2^{1 / 2}\right)^4 \\
b & =2^2=4 .
\end{aligned}
$$ and $$
\begin{aligned}
ab^{1/2} & =4 \\
a4^{1/2} & =4 \\
2a& =4 \\
a&= 2
\end{aligned}
$$ Hence $$
g(x)=2\cdot\left(4\right)^x
$$
