Answer
$f(x)=2(\frac{1}{3})^{x}$
Work Step by Step
The given curve in the question passing through the points $(0,2)$ and
$(2,\frac{2}{9})$.
Therefore,
$f(0)=Cb^{0}=2$
$f(2)=Cb^{2}=\frac{2}{9}$
Further,
$\frac{f(2)}{f(0)}=\frac{Cb^{2}}{Cb}=\frac{2/9}{2}$
This implies
$b^{2}=\frac{1}{9}$
$b=\frac{1}{3}$
Also,
$Cb^{2}=\frac{2}{9}$
Thus, $C=\frac{2/9}{b}$
Plug $\frac{1}{3}$ for $b$.
$C=2$
Hence, the required exponential function is $f(x)=2(\frac{1}{3})^{x}$.
