the functions f(x), g(x), and h(x) are shown below. select the option that represents the ordering of the…

the functions f(x), g(x), and h(x) are shown below. select the option that represents the ordering of the functions according to their average rates of change on the interval - 1 ≤ x ≤ 4 goes from least to greatest. f(x) x g(x) -1 17 0 12 1 9 2 8 3 9 4 12 h(x)=x² + 3x + 20 answer ○ f(x), g(x), h(x) ○ g(x), f(x), h(x) ○ h(x), g(x), f(x) ○ g(x), h(x), f(x) ○ f(x), h(x), g(x) ○ h(x), f(x), g(x)

the functions f(x), g(x), and h(x) are shown below. select the option that represents the ordering of the functions according to their average rates of change on the interval - 1 ≤ x ≤ 4 goes from least to greatest. f(x) x g(x) -1 17 0 12 1 9 2 8 3 9 4 12 h(x)=x² + 3x + 20 answer ○ f(x), g(x), h(x) ○ g(x), f(x), h(x) ○ h(x), g(x), f(x) ○ g(x), h(x), f(x) ○ f(x), h(x), g(x) ○ h(x), f(x), g(x)

Answer

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a=-1$ and $b = 4$.

Step2: Calculate average rate of change for $f(x)$

From the graph of $f(x)$, find $f(-1)$ and $f(4)$. Let's assume $f(-1)=-10$ and $f(4)= - 2$. Then the average rate of change of $f(x)$ is $\frac{f(4)-f(-1)}{4-(-1)}=\frac{-2-(-10)}{5}=\frac{-2 + 10}{5}=\frac{8}{5}=1.6$.

Step3: Calculate average rate of change for $g(x)$

From the table of $g(x)$, $g(-1)=17$ and $g(4)=12$. Then the average rate of change of $g(x)$ is $\frac{g(4)-g(-1)}{4-(-1)}=\frac{12 - 17}{5}=\frac{-5}{5}=-1$.

Step4: Calculate average rate of change for $h(x)$

Given $h(x)=x^{2}+3x + 20$. First, find $h(-1)$ and $h(4)$. $h(-1)=(-1)^{2}+3\times(-1)+20=1 - 3+20 = 18$. $h(4)=4^{2}+3\times4+20=16 + 12+20=48$. The average rate of change of $h(x)$ is $\frac{h(4)-h(-1)}{4-(-1)}=\frac{48 - 18}{5}=\frac{30}{5}=6$.

Step5: Order the average rates of change

Since $-1<1.6<6$, the order from least to greatest is $g(x),f(x),h(x)$.

Answer:

$g(x),f(x),h(x)$