Answer: The correct option is (C) [tex]-\dfrac{1}{3}.[/tex]
Step-by-step explanation: We are given a function f(x) defined as follows :
[tex]f(x)=\dfrac{4x-6}{x}.[/tex]
We are to find the average rate of change of f(x) over the interval [-3, 6].
We know that'
the average rate of change of a function p(x) over an interval [a, b] is given by
[tex]A_v=\dfrac{p(b)-p(a)}{b-a}.[/tex]
For the given function, we have
[tex]f(-3)=\dfrac{4\times(-3)-6}{-3}=\dfrac{-18}{-3}=6,\\\\\\ f(6)=\dfrac{4\times6-6}{6}=\dfrac{18}{6}=3.[/tex]
Therefore, the required average rate of change over the interval [-3, 6] will be
[tex]A_v=\dfrac{f(6)-f(-3)}{6-(-3)}=\dfrac{3-6}{6+3}=-\dfrac{3}{9}=-\dfrac{1}{3}.[/tex]
Thus, the required average rate of change over the interval [-3, 6] is [tex]-\dfrac{1}{3}.[/tex]
Option (C) is CORRECT.
