The order of the three functions in order from the fastest decreasing average rate of change to the slowest decreasing average rate of change on the interval [0, 3] is is g(x) >f(x) > h(x)
What is Function?
A function from a set X to a set Y assigns to each element of X exactly one element of Y.
What is Exponential function?
A function whose value is a constant raised to the power of the argument, especially the function where the constant is e.
What is average rate of change?
It is a measure of how much the function changed per unit, on average, over that interval.
Given,
[tex]f(x) = 16(\frac{1}{2})^{x}[/tex]
interval = [0,3]
[tex]f(0)= 16(\frac{1}{2})^{0} =16 \\f(3)= 16(\frac{1}{2})^{3} =2[/tex]
Average rate of change = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
Average rate of change= [tex]\frac{2-16}{3-0}=-4.67[/tex]
Consider the function g(x)
g(0)=21
g(3)=1
Average rate of change = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
Average rate of change =[tex]\frac{1-27}{3-0}=-8.67[/tex]
Consider the exponential function
at x=0 the exponential function h =4
at x=0 the exponential function h =-3
Average rate of change = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
Average rate of change =[tex]\frac{-3-4}{3-0}=-2.33[/tex]
Hence, the order of the three functions in order from the fastest decreasing average rate of change to the slowest decreasing average rate of change on the interval [0, 3] is g(x) >f(x) > h(x)
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