Answer
A) exponential
B) $f(x)=12.5\cdot (1.1)^x$
Work Step by Step
Given
$$
\begin{array}{c|c}
\hline x & f(x) \\
\hline 0 & 12.5 \\
1 & 13.75 \\
2 & 15.125 \\
3 & 16.638 \\
4 & 18.301 \\
\hline
\end{array}
$$
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.
$$
f(1)-f(0)=13.75-12.5=1.25
$$
$$
f(2)-f(1)=15.125-13.75=1.375
$$
Thus, the function is not linear. Check the ratios.
$$
\begin{aligned}
& \frac{f(1)}{f(0)}=\frac{13.75}{12.5}=1.1 \\
& \frac{f(2)}{f(1)}=\frac{15.25}{13.75}=1.1 .
\end{aligned}
$$
Hence, the function is exponential.
B) We know that
$$
f(x)=a b^x
$$
Thus,
$$
\begin{aligned}
& 12.5=f(0) \\
& 12.5=a b^0 \\
& 12.5=a
\end{aligned}
$$
and
$$
\begin{aligned}
& 13.75=f(1) \\
& 13.75=12.5 b\\
&b=\frac{13.75}{12.5}=1.1
\end{aligned}
$$
It follows that
$$
f(x)=12.5\cdot (1.1)^x
$$
