Answer
$f(t)=26-0.8 t$
$g(t)=30.4432 \cdot(0.9371)^t$
Work Step by Step
i) Let $f(t)=b+m t$, where $m$, the slope, is given by:
$$
m=\frac{\Delta f}{\Delta t}=\frac{f(25)-f(5)}{25-5}=\frac{6-22}{20}=-0.8 .
$$ and
$$
\begin{aligned}
f(5) & =b-(0.8) 5 \\
b & =f(5)+(0.8) 5=22+(0.8) 5=26
\end{aligned}
$$ Hence $$f(t)=26-0.8 t$$
ii) Let $g(t)=a b^t$. We know from the graph that $g(5)=a b^{5}=22$ and $g(25)=a b^{25}=6$. So
$$
\begin{aligned}
\frac{a b^{25}}{a b^{5}} & =\frac{6}{22} \\
b^{20} & =\frac{6}{22} \\
b & =\left(\frac{6}{22}\right)^{1 / 20}\approx 0.9371
\end{aligned}
$$ and $$
\begin{aligned}
& a b^{5}=22 \\
& a=b^{-5}\cdot 22 \\
&=\frac{22}{0.9371^5} \\
& =30.4432
\end{aligned}
$$ Hence $$g(t)= 30.4432\cdot\left(0.9371\right)^t$$
