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The problem asks to find the average rate of change of the function:

\[
f(x) = \frac{4}{-3x - 3}
\]

on the interval \(x = -4\) to \(x = 4\).

The average rate of change of a function \(f(x)\) on an interval \([a, b]\) is given by:

\[
\frac{f(b) - f(a)}{b - a}
\]

In this case, \(a = -4\) and \(b = 4\).

### Step 1: Calculate \(f(a)\), where \(a = -4\):
\[
f(-4) = \frac{4}{-3(-4) - 3} = \frac{4}{12 - 3} = \frac{4}{9}
\]

### Step 2: Calculate \(f(b)\), where \(b = 4\):
\[
f(4) = \frac{4}{-3(4) - 3} = \frac{4}{-12 - 3} = \frac{4}{-15} = -\frac{4}{15}
\]

### Step 3: Apply the formula for average rate of change:
\[
\text{Average rate of change} = \frac{f(4) - f(-4)}{4 - (-4)} = \frac{-\frac{4}{15} - \frac{4}{9}}{4 + 4}
\]

Simplifying the numerator:

\[
-\frac{4}{15} - \frac{4}{9} = -\left(\frac{4}{15} + \frac{4}{9}\right)
\]

Find a common denominator:

\[
\frac{4}{15} + \frac{4}{9} = \frac{4 \times 3}{45} + \frac{4 \times 5}{45} = \frac{12}{45} + \frac{20}{45} = \frac{32}{45}
\]

Thus:

\[
-\frac{4}{15} - \frac{4}{9} = -\frac{32}{45}
\]

Now divide by 8 (since \(4 - (-4) = 8\)):

\[
\text{Average rate of change} = \frac{-\frac{32}{45}}{8} = -\frac{32}{45} \times \frac{1}{8} = -\frac{32}{360} = -\frac{8}{90} = -\frac{4}{45}
\]

So, the average rate of change of the function on the interval \([-4, 4]\) is:

\[
-\frac{4}{45}
\]

Let me know if you would like further details or clarification!

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Here are 5 related questions:
1. What is the average rate of change of a linear function?
2. How do you interpret the result of the average rate of change in real-world problems?
3. How is the average rate of change different from the instantaneous rate of change?
4. Can the average rate of change be zero for non-constant functions?
5. How does the sign of the average rate of change inform us about the behavior of the function?

**Tip**: To find the average rate of change of a function, think of it as finding the slope of the secant line between two points on the curve!

Find the average rate of change of the function f(x) = 4 / (-3x - 3) on the interval from x = -4 to x = 4.

Find the average rate of change for the rational function f(x) = 4 / (-3x - 3) on the interval from x = -4 to x = 4. Detailed steps include applying the average rate of change formula, calculating values of the function, and simplifying the result.

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