Answer
(a) $5$
(b) $y=5x-2$
Work Step by Step
(a) Given $f(x)=5x-2$, we have the average rate of change
$R=\dfrac{f(x_2)-f(x_1)}{x_2-x_1}$
From $x_1=1$ to $x_2=3$, we have:
$R=\dfrac{f(3)-f(1)}{x_2-x_1}=\dfrac{5(3)-2-[5(1)-2]}{3-1}=5$
(b) Determine the value of the function at $x=1$ and $x=3$:
$f(1)=5(1)-2=3$
$f(3)=5(3)-2=13$
Thus, the two points are $(1,3),(3,13)$.
Find the slope $m$ of the line using the formula $m=\dfrac{y_2-y_1}{x_2-x_1}$:
$$m=\dfrac{13-3}{3-1}=5$$
Hence, and the equation of the line is:
$$\begin{align*}
y-3&=5(x-1)\\
y-3&=5x-5\\
y&=5x-5+3\\
y&=5x-2\end{align*}$$
