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.

The functions fx gx and hx are shown below Select the option that represents the ordering of the functions according to their average rates of change on the int class=
The functions fx gx and hx are shown below Select the option that represents the ordering of the functions according to their average rates of change on the int class=
The functions fx gx and hx are shown below Select the option that represents the ordering of the functions according to their average rates of change on the int class=

Respuesta :

SOLUTION

Given the question in the image, the following are the solution steps to answer the question.

STEP 1: Write the formula for the rate of change

[tex]\begin{gathered} rate\text{ of change}=\frac{f(b)-f(a)}{b-a} \\ rate\text{ of change}=\frac{y_2-y_1}{x_2-x_1} \end{gathered}[/tex]

STEP 2: Write the given intervals

[tex]-1\leq x\leq4[/tex]

STEP 3: Find the average rate of change of f(x)

[tex]\begin{gathered} Picking\text{ two points on the graphs, we have:} \\ (x_1,y_1)=\left(-1,5\right) \\ (x_2,y_2)=(4,0) \end{gathered}[/tex]

We substitute the coordinates into the rate of change formula:

[tex]rate\text{ of change}=\frac{0-5}{4-(-1)}=-\frac{5}{5}=-1[/tex]

STEP 4: Find the rate of change of g(x)

[tex]\begin{gathered} (x_1,y_1)=(-1,17) \\ (x_2,y_2)=(4,2) \\ rate\text{ of change}=\frac{2-17}{4-(-1)}=\frac{-15}{5}=-3 \end{gathered}[/tex]

STEP 5: Find the rate of change of h(x)

[tex]\begin{gathered} h(x)=-x^2-5x+37 \\ x_1=-1 \\ h(-1)=-(-1^2)-5(-1)+37=-1+5+37=41 \\ x_2=4 \\ h(4)=-(4^2)-5(4)+37=-16-20+37=1 \\ The\text{ new points become:} \\ (x_1,y_1)=(-1,41) \\ (x_2,y_2)=(4,1) \\ Average\text{ rate of change:} \\ \frac{1-41}{4-(-1)}=\frac{-40}{5}=-8 \end{gathered}[/tex]

STEP 6: Write the average rates of change for the functions

[tex]\begin{gathered} f(x)=-1 \\ g(x)=-3 \\ h(x)=-8 \end{gathered}[/tex]

The average rates of change in ascending order will be -8,-3,-1

Hence, the arrangement of the functions according to their ascending order of average rates of changes are:

[tex]h(x),g(x),f(x)[/tex]

ACCESS MORE
EDU ACCESS
Universidad de Mexico