Question

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 3≤x≤4 goes from least to greatest.

Answer 1

The rate of change, m, of a function f is;

`m=(f(b)-f(a))/(b-a)`

Given the graph, f(x);

Its rate of change over the given interval is;

`\begin{gathered} m=(f(4)-f(3))/(4-3) \\ \\ m=(6-(-2))/(1) \\ \\ m=8 \end{gathered}`

Given the table of g(x);

Its rate of change over the interval is;

`\begin{gathered} m=(g(4)-g(3))/(4-3) \\ \\ m=(13-18)/(1) \\ \\ m=-5 \end{gathered}`

Also, given the function of h(x);

`\begin{gathered} h(x)=-x^2+3x+21 \\ \\ h(4)=-(4)^2+3(4)+21 \\ \\ h(4)=-16+12+21 \\ \\ h(4)=17 \\ \\ h(3)=-(3)^2+3(3)+21 \\ \\ h(3)=-9+9+21 \\ \\ h(3)=21 \end{gathered}`

Its rate of change over the interval is;

`\begin{gathered} m=(17-21)/(4-3) \\ \\ m=-(4)/(1) \\ \\ m=-4 \end{gathered}`

The order of the rate of change from the least to the greatest is;

`-5,-4,8`

CORRECT OPTION:

`g(x),h(x),f(x)`