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Let's break down the two parts of the question based on the table provided in your image.

### Part a. What is the value of \( f(10) \)?

From the table, you can directly read off the value of \( f(10) \).

- When \( x = 10 \), \( f(10) = 1.1 \).

### Part b. Calculate the average rate of change from \( x = 5 \) to \( x = 10 \). Use your answer to find \( f(7) \).

The average rate of change between two points is calculated using the formula:

\[
\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

For \( x_1 = 5 \) and \( x_2 = 10 \):

\[
\text{Average rate of change} = \frac{f(10) - f(5)}{10 - 5}
\]

Substitute the values from the table:

\[
f(10) = 1.1 \quad \text{and} \quad f(5) = 4.3
\]

Now calculate:

\[
\text{Average rate of change} = \frac{1.1 - 4.3}{10 - 5} = \frac{-3.2}{5} = -0.64
\]

So, the average rate of change is \( -0.64 \).

### Use this to estimate \( f(7) \).

Since 7 is between 5 and 10, we can assume a linear change between those points. To find \( f(7) \), we'll use the slope \( m = -0.64 \) and apply the point-slope form of the line between \( x = 5 \) and \( x = 10 \). Let's take the point \( (5, 4.3) \):

\[
f(7) = f(5) + m(x - 5)
\]

Substitute \( x = 7 \), \( f(5) = 4.3 \), and \( m = -0.64 \):

\[
f(7) = 4.3 + (-0.64)(7 - 5) = 4.3 + (-0.64)(2) = 4.3 - 1.28 = 3.02
\]

So, \( f(7) \approx 3.02 \).

### Summary:

- \( f(10) = 1.1 \)
- Average rate of change between \( x = 5 \) and \( x = 10 \) is \( -0.64 \)
- \( f(7) \approx 3.02 \)

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Would you like any additional details on the process?

Here are 5 related questions to further your understanding:

1. How would the calculation change if the points between 5 and 10 were not linear?
2. How can you use the concept of the average rate of change in real-world problems?
3. Can you calculate the rate of change between \( x = 0 \) and \( x = 5 \)?
4. How does the slope of a function inform us about its behavior over an interval?
5. What would \( f(12) \) be assuming the same average rate of change applies beyond \( x = 10 \)?

**Tip**: When dealing with linear approximations, the assumption is that the rate of change remains constant within the interval, but this might not always be true for non-linear functions!

The following is a partial table of values for f(x). Find the following:
1. What is the value of f(10)?
2. Calculate the average rate of change from x = 5 to x = 10. Use your answer to find f(7).

Understand how to find the value of f(10) from a function table, calculate the average rate of change from x = 5 to x = 10, and use linear approximation to estimate f(7). This detailed guide includes step-by-step solutions and explanations, suitable for students in Grades 9-10.

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