Question

Given the function g(x) = 6(4)x, Section A is from x = 0 to x = 1 and Section B is from x = 2 to x = 3.

Part A: Find the average rate of change of each section. (4 points)

Part B: How many times greater is the average rate of change of Section B than Section A? Explain why one rate of change is greater than the other. (6 points)

Answer 1

`\text{A function is a relationship or expression that involves}`
`\text{one or more variables.}`


`\text{When we are given Composed Functions), we have the ability}`
`\text{to combine them in such a way that the}`
`\text{outcomes of one function becomes the other}`


`\text{For example - }`

If
`f(x)=3x-1` and
`g(x)=x^3+2`, then what is
`f(g(3))`?


`g(x)=x^3+2`


`g(x)=(3)^3+2`


`= 29`


`\text{Since}`
`g(3) = 29` then
`f(g(3))=f(29)`


`\text{Now let's evaluate}`
`f=(29)`


`f(x) = 3x-1`


`f(29)=3(29)-1`


`=86`


`f(g(3))=f(29)=86`


`\text{To find the value of g(x) we need to substitute}`
`\text{a number into the function's formula: }`


`g\left(x\right)=6\left(4\right)x`


`(d)/(dx) (6*4x)`


`6*4(d)/(dx)(x)`


`6*4*1`


`= 24`


`\text{The`


`\text{Formula:}`
`\frac{\text{(change in f(x) }}{\text{(change in 'x')}}`


`\text{In Section A:}`


`\text{Length of section}` = (1 - 0) = 1


`f(1) = 6(4)x`


`f(0) = 0`


`\text{Change in the value of the given function}` =
`(24 - 0) = 24`


`\frac{\text{ (change in the value of the function)}}{\text{(size of the section)}}`
`(24)/(1) = 24`