To find the average rate of change of a function over an interval, we use the formula:

$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

where $a$ and $b$ are the endpoints of the interval for $x$, and $f(a)$ and $f(b)$ are the corresponding function values.

In this problem:

• The interval is $24 \leq x \leq 96$.
• From the table, $f(24) = 50$ and $f(96) = 38$.

Now, substitute these values into the formula:

$\text{Average Rate of Change} = \frac{f(96) - f(24)}{96 - 24} = \frac{38 - 50}{96 - 24} = \frac{-12}{72} = -\frac{1}{6}$

So, the average rate of change of the function over the interval $24 \leq x \leq 96$ is:

$-\frac{1}{6}$

Would you like more details on this calculation, or do you have further questions?

Here are five related questions to consider:

1. How does the average rate of change differ from the instantaneous rate of change?
2. What does a negative rate of change indicate about a function?
3. How can this method be applied to different intervals within the table?
4. How would the calculation change if $f(x)$ values had been increasing over the interval?
5. Why is it helpful to use tables to find average rates of change in certain contexts?

Tip: When finding the average rate of change, always ensure the values for $f(a)$ and $f(b)$ correspond to the exact interval specified to avoid errors.