To solve the problem of finding the average rate of change (ARC) of the function $f(x) = 9x - 8$ over the interval $[2,3]$, we can use the formula for the average rate of change:

$\text{ARC} = \frac{f(b) - f(a)}{b - a}$ where $a = 2$ and $b = 3$.

Step-by-Step Solution:

1. First, calculate $f(2)$: $f(2) = 9(2) - 8 = 18 - 8 = 10$

2. Now, calculate $f(3)$: $f(3) = 9(3) - 8 = 27 - 8 = 19$

3. Apply these values to the ARC formula: $\text{ARC} = \frac{f(3) - f(2)}{3 - 2} = \frac{19 - 10}{1} = 9$

Thus, the average rate of change is $9$, which is not listed as an option in the problem (the choices are $10$, $6$, and $12$). This suggests the problem might have a typo or incorrect answer choices.

Would you like further clarification or details on the steps?

Here are 5 related questions to expand your understanding:

1. What is the geometric meaning of the average rate of change?
2. How does the ARC formula relate to the derivative of a function?
3. Can the average rate of change be negative? Under what conditions?
4. What is the significance of linearity in functions like $f(x) = 9x - 8$ when calculating the ARC?
5. How would the ARC formula change if the function was non-linear, such as a quadratic function?

Tip: When dealing with linear functions, the average rate of change over any interval is always the slope of the line.