The Solution to Question 16C:
Given the functions:
[tex]g(x)=\begin{cases}\frac{x}{3}+2\text{ if x<1} \\ \\ 4x-2\text{ if x}\ge1\end{cases}[/tex][tex]f(x)=2x+7_{}[/tex]We are required to investigate which of the two functions has the greater average rate of change over the interval
[tex]-12\leq x\leq8[/tex]Step 1:
The formula for the rate of change is:
[tex]\begin{gathered} \frac{f(b)-f(a)}{b-a}\text{ if b>a} \\ \\ a=-12 \\ b=8 \end{gathered}[/tex]Step 2:
We shall find the rate of change for f(x) and g(x).
The rate of change for g(x):
[tex]=\frac{(32-2)-(-4+2)}{8+12}=\frac{30--2}{20}=\frac{32}{20}=1.6[/tex]Similarly,
The rate of change for f(x):
[tex]\text{ Rate of change=}\frac{f(8)-f(-12)}{8--12}=\frac{\lbrack2(8)+7\rbrack-\lbrack2(-12)+7\rbrack}{8+12}[/tex][tex]=\frac{(16+7)-(-24+7)}{20}=\frac{23--17}{20}=\frac{23+17}{20}=\frac{40}{20}=2[/tex]Comparing the rate of change of both functions, we have that:
[tex]\begin{gathered} g(x)\text{ rate of change = 1.6 while that of f(x) =2} \\ \text{ since 2>1.6, we conclude that f(x) has a greater rate of change.} \end{gathered}[/tex]Thus, the function f(x) has a greater average rate of change than the function g(x) since 2 is greater than 1.6 over the given interval.
