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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.

Given a table for f(x) and g(x), find the average rate of change of g(x) from x = 4 to x = 6 and f(x) from x = 6 to x = 7.

Learn how to find the average rate of change for the functions g(x) from x = 4 to x = 6 and f(x) from x = 6 to x = 7 using the slope formula. This example demonstrates the key concepts of functions and linear approximation for students in Grades 8-10.

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