f(x)=x-6/x+9 find the average rate of change of f on [x,x+h]

To find the average rate of change of the function \( f(x) = \frac{x - 6}{x + 9} \) on the interval \([x, x+h]\), we use the following formula for the average rate of change of a function between two points:

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

### Step 1: Find \( f(x+h) \)

Substitute \( x+h \) into the function \( f(x) = \frac{x - 6}{x + 9} \):

\[
f(x+h) = \frac{(x+h) - 6}{(x+h) + 9} = \frac{x + h - 6}{x + h + 9}
\]

### Step 2: Compute the difference \( f(x+h) - f(x) \)

Now, compute the difference between \( f(x+h) \) and \( f(x) \):

\[
f(x+h) - f(x) = \frac{x + h - 6}{x + h + 9} - \frac{x - 6}{x + 9}
\]

To subtract these two fractions, we'll need a common denominator, which is \( (x+h+9)(x+9) \):

\[
f(x+h) - f(x) = \frac{(x + h - 6)(x + 9) - (x - 6)(x + h + 9)}{(x+h+9)(x+9)}
\]

### Step 3: Simplify the numerator

Now expand and simplify the numerator:

\[
(x + h - 6)(x + 9) = x^2 + hx - 6x + 9h - 54
\]
\[
(x - 6)(x + h + 9) = x^2 + hx - 6x + 9x - 6h - 54
\]

Now subtract the two expressions:

\[
(x^2 + hx - 6x + 9h - 54) - (x^2 + hx + 3x - 6h - 54)
\]

Simplify this:

\[
= (x^2 - x^2) + (hx - hx) + (-6x - 3x) + (9h - (-6h)) + (-54 + 54)
\]
\[
= -9x + 15h
\]

### Step 4: Divide by \( h \)

Now, divide this by \( h \), the difference \( (x+h) - x = h \):

\[
\frac{f(x+h) - f(x)}{h} = \frac{-9x + 15h}{h(x+h+9)(x+9)}
\]

Simplify the fraction:

\[
= \frac{-9x + 15h}{h(x+h+9)(x+9)} = \frac{-9(x - \frac{5h}{3})}{(x+h+9)(x+9)}
\]

Thus, the average rate of change of \( f(x) = \frac{x - 6}{x + 9} \) on the interval \([x, x+h]\) is:

\[
\boxed{\frac{-9x + 15h}{h(x+h+9)(x+9)}}
\]

Would you like more details on any step or a different approach?

Here are some related questions you might find interesting:
1. How does the average rate of change differ from the instantaneous rate of change?
2. What is the derivative of \( f(x) = \frac{x - 6}{x + 9} \)?
3. How would the answer change if the function was \( f(x) = \frac{x + 6}{x - 9} \)?
4. What is the significance of the common denominator in this computation?
5. How would this concept apply to real-world rate of change problems?

**Tip**: To solve problems involving rates of change, breaking down the equation step-by-step helps ensure accuracy and reduces complexity when simplifying expressions.

Discover how to calculate the average rate of change for the rational function f(x) = (x - 6) / (x + 9) over the interval [x, x+h]. This detailed solution covers essential algebraic concepts and the difference quotient, ideal for students in Grades 10-12.

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