The image shows a table for the functions $f(x)$ and $g(x)$, along with a request to find the average rates of change for both functions over specific intervals:

1. Average rate of change of $g(x)$ from $x = 4$ to $x = 6$: The average rate of change formula is given by: $\text{Average rate of change} = \frac{g(x_2) - g(x_1)}{x_2 - x_1}$ For this case: $\frac{g(6) - g(4)}{6 - 4}$

2. Average rate of change of $f(x)$ from $x = 6$ to $x = 7$: Similarly, the formula applies: $\frac{f(7) - f(6)}{7 - 6}$

Let's extract the corresponding values from the table and compute these.

From the table:

• $g(6) = 5$, $g(4) = 4$
• $f(7) = 4$, $f(6) = 8$

Now let's compute these average rates of change.The average rates of change are as follows:

1. The average rate of change of $g(x)$ from $x = 4$ to $x = 6$ is $0.5$.
2. The average rate of change of $f(x)$ from $x = 6$ to $x = 7$ is $-4.0$.

Would you like more details on how these values were calculated, or do you have any further questions?

Here are 5 related questions to expand on the topic:

1. What is the geometric interpretation of the average rate of change?
2. How does the average rate of change relate to the concept of a derivative?
3. Can the average rate of change be used to estimate instantaneous rates of change?
4. What does a positive or negative average rate of change indicate about the behavior of the function?
5. How does the average rate of change differ across linear and nonlinear functions?

Tip: The average rate of change formula is similar to the slope formula for a straight line, but it applies to any two points on a function.