Avg rate of change is 2 on the interval -1≤ 𝑥 ≤ 1.
Avg rate of change is 7 on the interval 1 ≤ 𝑥 ≤3.
What is the average rate of change of 𝑔 on the interval
-3 ≤𝑥≤-1 ?

To solve this problem, we need to use the information given and understand how the average rate of change (AROC) is calculated. The average rate of change of a function \( g(x) \) over an interval \([a, b]\) is given by:

\[
\text{AROC} = \frac{g(b) - g(a)}{b - a}
\]

### Step 1: Interpret the given data

We know the following:

1. The average rate of change on the interval \([-1 \leq x \leq 1]\) is 2.
   
   \[
   \frac{g(1) - g(-1)}{1 - (-1)} = 2
   \]

2. The average rate of change on the interval \([1 \leq x \leq 3]\) is 7.

   \[
   \frac{g(3) - g(1)}{3 - 1} = 7
   \]

### Step 2: Solve for \( g(1) - g(-1) \) and \( g(3) - g(1) \)

- From the first interval:

\[
\frac{g(1) - g(-1)}{2} = 2 \quad \Rightarrow \quad g(1) - g(-1) = 4
\]

- From the second interval:

\[
\frac{g(3) - g(1)}{2} = 7 \quad \Rightarrow \quad g(3) - g(1) = 14
\]

### Step 3: Use the above results

To find the average rate of change of \( g(x) \) on the interval \([-3 \leq x \leq -1]\), we need information about \( g(-3) \) and \( g(-1) \). However, since we don’t have direct values for these points, and they are not connected to the intervals given, we cannot determine the average rate of change for \([-3, -1]\) without additional information about the function on that interval.

### Conclusion

Without more details about \( g(x) \) on the interval \([-3, -1]\), it is impossible to determine the average rate of change on this interval. If additional data is provided, the solution can be calculated accordingly.

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Do you want more details, or do you have any further questions?

Here are 5 related questions:

1. How do you find the average rate of change of a function over a given interval?
2. What is the significance of the average rate of change in calculus?
3. How does the average rate of change relate to the derivative of a function?
4. Can the average rate of change be used to estimate function behavior over larger intervals?
5. How would you approach finding the function \( g(x) \) from the given information?

**Tip:** The average rate of change is equivalent to the slope of the secant line connecting two points on the function’s graph.

Learn how to calculate the average rate of change of a function over specified intervals with a step-by-step solution. This problem involves interpreting given data and using the AROC formula. Suitable for high school students studying calculus.

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