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The problem asks to find the average rate of change of the function \( g(x) = \frac{1}{x + 3} \) over the interval \([5, 5 + h]\), where \( h \neq 0 \).

### Step-by-Step Solution:

The average rate of change of a function \( g(x) \) over an interval \([a, b]\) is given by the formula:
\[
\text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a}
\]
Here, \( a = 5 \) and \( b = 5 + h \).

1. **Evaluate \( g(a) = g(5) \):**
   \[
   g(5) = \frac{1}{5 + 3} = \frac{1}{8}
   \]

2. **Evaluate \( g(b) = g(5 + h) \):**
   \[
   g(5 + h) = \frac{1}{(5 + h) + 3} = \frac{1}{h + 8}
   \]

3. **Substitute into the average rate of change formula:**
   \[
   \text{Average Rate of Change} = \frac{\frac{1}{h + 8} - \frac{1}{8}}{(5 + h) - 5}
   \]
   Simplify the denominator:
   \[
   (5 + h) - 5 = h
   \]
   So, the expression becomes:
   \[
   \text{Average Rate of Change} = \frac{\frac{1}{h + 8} - \frac{1}{8}}{h}
   \]

4. **Simplify the numerator:**
   The difference of fractions in the numerator is:
   \[
   \frac{1}{h + 8} - \frac{1}{8} = \frac{8 - (h + 8)}{8(h + 8)} = \frac{8 - h - 8}{8(h + 8)} = \frac{-h}{8(h + 8)}
   \]

5. **Substitute back into the expression:**
   \[
   \text{Average Rate of Change} = \frac{\frac{-h}{8(h + 8)}}{h}
   \]
   Simplify this by dividing by \( h \):
   \[
   \text{Average Rate of Change} = \frac{-h}{8(h + 8)} \times \frac{1}{h} = \frac{-1}{8(h + 8)}
   \]

### Final Answer:
The average rate of change is:
\[
\boxed{\frac{-1}{8(h + 8)}}
\]

Would you like further clarification or a breakdown of any specific part?

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Here are 5 related questions:
1. How do you interpret the average rate of change geometrically?
2. What is the derivative of \( g(x) = \frac{1}{x+3} \)?
3. How does the average rate of change compare to the instantaneous rate of change?
4. What happens to the average rate of change as \( h \) approaches 0?
5. How would the formula change if the function had a different denominator, like \( g(x) = \frac{1}{x + 5} \)?

**Tip:** The average rate of change formula is the same as the slope of the secant line connecting two points on the graph.

Find the average rate of change for the function g(x) = 1 / (x + 3) over the interval [5, 5 + h], where h ≠ 0. Simplify your answer.

Learn how to calculate the average rate of change for the rational function g(x) = 1 / (x + 3) over a given interval [5, 5 + h]. This step-by-step solution simplifies the problem using algebraic methods, perfect for high school students studying rate of change.

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