Answer
$
f(x)=2\cdot\left(\frac{3}{2}\right)^x
$
Work Step by Step
We want to have $f(x)=a b^{x}$.
The graph shows that the points $(-2,8/9),(2,9/2)$ lie on the curve of $f(x)$. We must have $f(-2)=a b^{-2}= 8/9$ and $f(2)=a b^{2}= 9/2$. This gives
$$
\begin{aligned}
\frac{a b^{2}}{a b^{-2}} & =\frac{f(2)}{f(-2)} \\
b^{4} & =\frac{9/2}{8/9}= \frac{81}{16} \\
b & =\left(\frac{81}{16}\right)^{1 / 4} = \frac{3}{2}
\end{aligned}
$$ We determine $a$:
$$
\begin{aligned}
ab^{-2}& =\frac{8}{9} \\
a & =b^2\cdot\frac{8}{9} \\
a & =\frac{9}{4}\cdot \frac{8}{9} =2
\end{aligned}
$$ It follows that $$
f(x)=2\cdot\left(\frac{3}{2}\right)^x
$$
