To find the average rate of change of the function $g(x) = \frac{1}{-x}$ over the interval $[-2, 8]$, we use the formula:

$\text{Average rate of change} = \frac{g(b) - g(a)}{b - a}$

where $a = -2$ and $b = 8$.

1. First, calculate $g(-2)$:

$g(-2) = \frac{1}{-(-2)} = \frac{1}{2}$

2. Next, calculate $g(8)$:

$g(8) = \frac{1}{-8} = -\frac{1}{8}$

3. Now, substitute these values into the formula for the average rate of change:

$\text{Average rate of change} = \frac{g(8) - g(-2)}{8 - (-2)} = \frac{-\frac{1}{8} - \frac{1}{2}}{10}$

4. Simplify the expression:

$\text{Average rate of change} = \frac{-\frac{1}{8} - \frac{4}{8}}{10} = \frac{-\frac{5}{8}}{10} = -\frac{5}{80} = -\frac{1}{16}$

So, the average rate of change of the function $g(x) = \frac{1}{-x}$ over the interval $[-2, 8]$ is:

$\boxed{-\frac{1}{16}}$

Would you like more details or have any questions?

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Here are some related questions:

1. What does a negative average rate of change imply about the behavior of the function over the interval?
2. How would the rate of change differ if the interval was [-2, 2] instead of [-2, 8]?
3. What is the relationship between the average rate of change and the slope of the secant line connecting two points on the curve?
4. How does the function $g(x) = \frac{1}{-x}$ behave as $x$ approaches zero?
5. How can the concept of average rate of change be applied to analyze the speed of a moving object?

Tip: The average rate of change can help you understand whether a function is increasing or decreasing over an interval, and it can provide insights into the overall behavior of the function in different contexts.