Answer
B) $
f(x)=2.23\cdot(1.032)^x
$
Work Step by Step
A)
We know that a function is linear if the differences of successive values of the function is a constant. A function is exponential if the ratios of successive values of the function is a constant.
$$
\begin{aligned}
& f(3)-f(0)=2.45-2.23=0.22\\
& f(6)-f(3)=2.7-2.45=0.47
\end{aligned}
$$
The function is not linear. The function $h$, is linear for the same reason.
$$
\begin{aligned}
& \frac{f(3)}{f(0)}=\frac{2.45}{2.23}=1.1 \\
& \frac{f(6)}{f(3)}=\frac{2.7}{2.45}=1.1
\end{aligned}
$$ Hence, the function is exponential.
B) We want to have $f(x)=a b^x$.
The table shows that the points $(0,2.23),(3,2.45)$ lie on the curve of $f(x)$. We must have $f(0)=a b^0=a=2.23$ and so $f(x)=2.23 b^x$. $f(3)=2.23 b^3=2.45 \Longrightarrow b=(2.45/2.23)^{1 / 3}=1.032$
Hence:
$$
f(x)=2.23\cdot(1.032)^x
$$
