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

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = k(x)$ on the interval $a\leq x\leq b$ is $\frac{k(b)-k(a)}{b - a}$. Here, $a = 3$ and $b = 4$.

Step2: Calculate average rate of change of $f(x)$ from the graph

From the graph of $f(x)$, when $x = 3$, $f(3)\approx - 2$ and when $x = 4$, $f(4)\approx4$. The average rate of change of $f(x)$ is $\frac{f(4)-f(3)}{4 - 3}=\frac{4-(-2)}{1}=6$.

Step3: Calculate average rate of change of $g(x)$ from the table

From the table of $g(x)$, $g(3)=9$ and $g(4)=6$. The average rate of change of $g(x)$ is $\frac{g(4)-g(3)}{4 - 3}=\frac{6 - 9}{1}=-3$.

Step4: Calculate average rate of change of $h(x)$ using the formula

Given $h(x)=-x^{2}+3x + 14$. Then $h(3)=-3^{2}+3\times3 + 14=14$ and $h(4)=-4^{2}+3\times4 + 14=-16 + 12+14 = 10$. The average rate of change of $h(x)$ is $\frac{h(4)-h(3)}{4 - 3}=\frac{10 - 14}{1}=-4$.

Step5: Order the average rates of change

Comparing the average rates of change: $-4<-3<6$, so $h(x),g(x),f(x)$.

Answer:

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