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

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

where $f(a)$ and $f(b)$ are the function values at the endpoints of the interval $[a, b]$.

a) Average rate of change of $g(x)$ from $x = 6$ to $x = 7$:

1. From the table:

   • $g(6) = 9$
   • $g(7) = 0$

2. Apply the formula: $\text{Average Rate of Change of } g(x) = \frac{g(7) - g(6)}{7 - 6} = \frac{0 - 9}{1} = \frac{-9}{1} = -9$

So, the average rate of change of $g(x)$ from $x = 6$ to $x = 7$ is $-9$.

b) Average rate of change of $f(x)$ from $x = 2$ to $x = 6$:

1. From the table:

   • $f(2) = 6$
   • $f(6) = 0$

2. Apply the formula: $\text{Average Rate of Change of } f(x) = \frac{f(6) - f(2)}{6 - 2} = \frac{0 - 6}{4} = \frac{-6}{4} = -1.5$

So, the average rate of change of $f(x)$ from $x = 2$ to $x = 6$ is $-1.5$.

Would you like more details on these calculations, or do you have any further questions?

Here are 5 related questions you might find useful:

1. What is the significance of the average rate of change in a function?
2. How does the average rate of change relate to the slope of a line?
3. Can you explain how the rate of change of a linear function differs from that of a non-linear function?
4. How would you calculate the average rate of change if given a continuous function instead of a table of values?
5. How can you determine if a function is increasing or decreasing based on the average rate of change?

Tip: The average rate of change between two points on a function is the slope of the line that connects those points. This concept is crucial when analyzing linearity and trends in data.