for f(x)=(x-6)/(x+9) construct and simplify the difference quotient

To construct and simplify the difference quotient for the function \( f(x) = \frac{x-6}{x+9} \), we follow these steps:

1. **Difference Quotient Formula**: The difference quotient for a function \( f(x) \) is given by:
   \[
   \frac{f(x+h) - f(x)}{h}
   \]

2. **Substitute \( f(x) \) and \( f(x+h) \) into the formula**:
   \[
   f(x) = \frac{x-6}{x+9}
   \]
   \[
   f(x+h) = \frac{(x+h)-6}{(x+h)+9} = \frac{x+h-6}{x+h+9}
   \]

3. **Construct the difference quotient**:
   \[
   \frac{f(x+h) - f(x)}{h} = \frac{\frac{x+h-6}{x+h+9} - \frac{x-6}{x+9}}{h}
   \]

4. **Simplify the numerator**:
   \[
   \frac{x+h-6}{x+h+9} - \frac{x-6}{x+9}
   \]
   To combine these fractions, find a common denominator:
   \[
   \frac{(x+h-6)(x+9) - (x-6)(x+h+9)}{(x+h+9)(x+9)}
   \]
   
5. **Expand and combine the terms in the numerator**:
   \[
   \frac{(x+h-6)(x+9) - (x-6)(x+h+9)}{(x+h+9)(x+9)}
   \]
   \[
   = \frac{(x^2 + 9x + hx + 9h - 6x - 54) - (x^2 + hx + 9x + 9h - 6x - 54)}{(x+h+9)(x+9)}
   \]
   Simplifying the expanded terms:
   \[
   = \frac{(x^2 + 9x + hx + 9h - 6x - 54) - (x^2 + hx + 9x + 9h - 6x - 54)}{(x+h+9)(x+9)}
   \]
   Notice that all the terms cancel each other out in the numerator:
   \[
   = \frac{(x^2 + 9x + hx + 9h - 6x - 54) - (x^2 + hx + 9x + 9h - 6x - 54)}{(x+h+9)(x+9)}
   \]
   \[
   = \frac{0}{(x+h+9)(x+9)}
   \]
   This leads to:
   \[
   = \frac{h(x-6)(x+9) - h(x-6)(x+h+9)}{h(x+h+9)(x+9)}
   \]
   Factor out \( h \) in the numerator:
   \[
   = \frac{h \left[ (x-6)(x+9) - (x-6)(x+h+9) \right]}{h(x+h+9)(x+9)}
   \]
   Simplify the numerator further:
   \[
   = \frac{h \left[ (x-6)(x+9) - (x-6)(x+h+9) \right]}{h(x+h+9)(x+9)}
   \]
   \[
   = \frac{h(x-6)(x+9-x-h-9)}{h(x+h+9)(x+9)}
   \]
   \[
   = \frac{h(x-6)(-h)}{h(x+h+9)(x+9)}
   \]
   \[
   = \frac{-h(x-6)}{h(x+h+9)(x+9)}
   \]

6. **Cancel the \( h \) term**:
   \[
   = \frac{-(x-6)}{(x+h+9)(x+9)}
   \]

Thus, the simplified difference quotient for \( f(x) = \frac{x-6}{x+9} \) is:
\[
\frac{-(x-6)}{(x+h+9)(x+9)}
\]

Would you like any further details or have any questions? Here are some related questions you might find helpful:

1. How do you find the derivative of a rational function?
2. What is the geometric interpretation of the difference quotient?
3. Can you provide examples of difference quotients for other types of functions?
4. How do you simplify complex rational expressions?
5. What is the importance of the difference quotient in calculus?

**Tip:** When simplifying the difference quotient, be meticulous with algebraic manipulations to avoid errors in combining and canceling terms.

Learn how to construct and simplify the difference quotient for the function f(x) = (x-6)/(x+9) with a detailed step-by-step solution. This problem covers key concepts in calculus, including the difference quotient formula for rational functions. Suitable for college-level students.

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