To find the average rate of change of a function $f$ over an interval $[a, b]$, where $a \neq b$, we use the formula:

$\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}$

This formula essentially measures the change in the function's value (i.e., $f(b) - f(a)$) over the change in the input (i.e., $b - a$).

Now, let's compare this with the given options:

1. $f(b) - f(a)$: This expression represents the change in the function values only and does not take into account the interval length $b - a$. This is not the average rate of change.

2. $\frac{f(b) - f(a)}{a - b}$: This expression is close, but the denominator is $a - b$, which is negative if $a < b$. The correct formula should have $b - a$ in the denominator. Thus, this is not the correct expression either.

3. $\frac{f(b) - f(a)}{b - a}$: This expression matches the formula for the average rate of change. It correctly accounts for the change in function value over the change in input.

4. None of the above: Since option (iii) matches the correct formula, this option is incorrect.

Therefore, the correct answer is:

(iii) $\frac{f(b) - f(a)}{b - a}$

Would you like more details or have any questions? Here are five related questions to consider:

1. How does the average rate of change relate to the derivative of a function?
2. What is the average rate of change for a linear function over any interval $[a, b]$?
3. How can the concept of average rate of change be applied to real-world problems?
4. What does the sign (positive/negative) of the average rate of change indicate about the function?
5. How does the average rate of change differ from the instantaneous rate of change?

Tip: The average rate of change of a function over an interval can be thought of as the slope of the secant line that passes through the points $(a, f(a))$ and $(b, f(b))$ on the function's graph.