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To find the average rate of change of the function \( f(x) = \frac{1}{x+5} \) over the interval \([4, 4+h]\), we will use the formula for the average rate of change:

\[
\text{Average Rate of Change} = \frac{f(4+h) - f(4)}{(4+h) - 4}.
\]

### Step 1: Find \( f(4+h) \)
Substitute \( x = 4 + h \) into the function \( f(x) \):
\[
f(4+h) = \frac{1}{(4+h) + 5} = \frac{1}{9 + h}.
\]

### Step 2: Find \( f(4) \)
Substitute \( x = 4 \) into the function:
\[
f(4) = \frac{1}{4 + 5} = \frac{1}{9}.
\]

### Step 3: Set up the formula for the average rate of change
Now, apply these values into the formula:
\[
\text{Average Rate of Change} = \frac{\frac{1}{9+h} - \frac{1}{9}}{(4+h) - 4}.
\]

This simplifies to:
\[
\text{Average Rate of Change} = \frac{\frac{1}{9+h} - \frac{1}{9}}{h}.
\]

### Step 4: Simplify the expression
To simplify the numerator, find a common denominator between \( \frac{1}{9+h} \) and \( \frac{1}{9} \):
\[
\frac{1}{9+h} - \frac{1}{9} = \frac{9 - (9+h)}{9(9+h)} = \frac{-h}{9(9+h)}.
\]

Thus, the expression becomes:
\[
\text{Average Rate of Change} = \frac{\frac{-h}{9(9+h)}}{h}.
\]

Cancel the \( h \) terms:
\[
\text{Average Rate of Change} = \frac{-1}{9(9+h)}.
\]

So, the average rate of change of \( f(x) \) on the interval \([4, 4+h]\) is:
\[
\boxed{\frac{-1}{9(9+h)}}.
\]

Would you like a deeper explanation or further details on any step?

Here are some related questions you might find interesting:
1. What is the difference between instantaneous and average rates of change?
2. How can you find the derivative of \( f(x) = \frac{1}{x+5} \)?
3. How does the value of \( h \) affect the average rate of change?
4. What happens to the average rate of change as \( h \) approaches 0?
5. Can this method be applied to other types of functions, like polynomials or trigonometric functions?

**Tip:** When simplifying expressions involving fractions, always try to find a common denominator first. This helps in combining the terms easily.

Given f(x) = 1 / (x + 5), find the average rate of change of f(x) on the interval [4, 4 + h]. Your answer will be an expression involving h.

This example explains how to find the average rate of change of f(x) = 1 / (x + 5) over the interval [4, 4+h]. It involves key concepts such as functions and algebraic manipulation to express the result as a function of h.

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