Answer:
The correct order of the functions from least to greatest according to the average rate of change on the interval from x = -1 to x = 3 is:
g(x) , f(x) , h(x)
( Since,
average rate of change of g(x) is: 1/2
average rate of change of f(x) is: 8
average rate of change of h(x) is: 12 )
Step-by-step explanation:
The average rate of a function from x=a to x=b is calculated by the formula:
\(\text{Average rate of change}=\dfrac{f(b)-f(a)}{b-a}\)
Here a= -1 and b=3
a)
The function f(x) is given by:
\(f(x)=(x+3)^2-2\)
\(f(-1)=(-1+3)^2-2\\\\i.e.\\\\f(-1)=2^2-2\\\\i.e.\\\\f(-1)=4-2\\\\i.e.\\\\f(-1)=2\)
\(f(3)=(3+3)^2-2\\\\i.e.\\\\f(3)=6^2-2\\\\i.e.\\\\f(3)=36-2\\\\i.e.\\\\f(3)=34\)
Hence, the average rate of change of f(x) is:
\(\text{Average rate of change}=\dfrac{34-2}{3-(-1)}\)
i.e.
\(\text{Average rate of change}=\dfrac{32}{4}\)
i.e.
\(\text{Average rate of change}=8\)
b)
The function g(x) is a straight line that passes through:
(-1,-2) and (3,0)
i.e.
g(-1)= -2
g(3)=0
i.e. the average rate of change is given by:
\(\text{Average rate of change}=\dfrac{g(3)-g(-1)}{3-(-1)}\)
i.e.
\(\text{Average rate of change}=\dfrac{0-(-2)}{4}\)
i.e.
\(\text{Average rate of change}=\dfrac{2}{4}\)
i.e.
\(\text{Average rate of change}=\dfrac{1}{2}\)
c)
Based on the table of values we have:
h(-1)= 14
and
h(3)= 62
\(\text{Average rate of change}=\dfrac{h(3)-h(-1)}{3-(-1)}\)
i.e.
\(\text{Average rate of change}=\dfrac{62-14}{4}\)
i.e.
\(\text{Average rate of change}=\dfrac{48}{4}\)
i.e.
\(\text{Average rate of change}=12\)
