Answer
$
f(x)=4\cdot\left(\frac{1}{10}\right)^x
$
Work Step by Step
We want to have $f(x)=a b^{x}$.
The graph shows that the points $(-2,400),(1,0.40)$ lie on the curve of $f(x)$.
We must have
$f(-2)=a b^{-2}= 400$ and $f(1)=a b^{1}= 0.40$. This gives
$$
\begin{aligned}
\frac{a b^{1}}{a b^{-2}} & =\frac{f(1)}{f(-2)} \\
b^{3} & =\frac{0.40}{400}= 0.001 \\
b & =\left(0.001\right)^{1 / 3} = \frac{1}{10}
\end{aligned}
$$ We determine $a$:
$$
\begin{aligned}
ab^{-2}& =400\\
a & =b^2\cdot\frac{8}{9} \\
a & =\frac{1}{100}\cdot 400=4
\end{aligned}
$$
It follows that $$
f(x)=4\cdot\left(\frac{1}{10}\right)^x
$$
