Question

How do you find the average rate of change for the function `f(x) = 3x - 2` on the indicated intervals [2,3]?

Answer 1

The average rate of change for `f(x)=3x-2` on the interval `[2,3]` is `3`

Explanation:

`f(x)=3x-2` is a linear function
`color(white)("XXXX")`with a slope (also called a "rate of change") of 3
`color(white)("XXXX")`for all point on the function line.

Since the rate of change for all points on the line is `3`
the average rate of change for any interval is also `3`

Answer 2

`(Deltaf)/(Deltax) = 3`

Explanation:

The average rate of change for a function `f(x)` between two points `x_1` and `x_2` is equal to

`color(blue)((Deltaf)/(Deltax) = (f(x_2) - f(x_1))/(x_2-x_1))`

In your case, you have

`f(x) = 3x-2` and `{(x_1 = 2), (x_2=3) :}`

Use the two values given to you to determine the values of `f(x)` in `x_1` and `x_2`

`f(x_1) = f(2) = 3 * 2 - 2 = 4`

`f(x_2) = f(3) = 3 * 3 - 2 = 7`

The average rate of change for your function between points `2` and `3` will thus be

`(Deltaf)/(Deltax) = (f(3) - f(2))/(3-2)`

`(Deltaf)/(Deltax) = (7 - 4)/1 = color(green)(3)`