difference quotient of f(x) = (x-1)/(x+2)

To find the **difference quotient** of the function \( f(x) = \frac{x - 1}{x + 2} \), we'll use the formula for the difference quotient:

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

### Step-by-step process:

1. **Find \( f(x+h) \):**
   Substitute \( x+h \) into \( f(x) \):
   \[
   f(x+h) = \frac{(x+h) - 1}{(x+h) + 2} = \frac{x+h-1}{x+h+2}
   \]

2. **Compute \( f(x+h) - f(x) \):**
   Now, subtract \( f(x) \) from \( f(x+h) \):
   \[
   f(x+h) - f(x) = \frac{x+h-1}{x+h+2} - \frac{x-1}{x+2}
   \]

   To subtract these fractions, first find a common denominator. The common denominator is \( (x+h+2)(x+2) \). So, we have:

   \[
   f(x+h) - f(x) = \frac{(x+h-1)(x+2) - (x-1)(x+h+2)}{(x+h+2)(x+2)}
   \]

   Expanding both numerators:
   \[
   (x+h-1)(x+2) = x^2 + 2x + hx + 2h - x - 2
   \]
   Simplifying:
   \[
   = x^2 + hx + (2h - 1)x - 2
   \]

   Similarly, expand \( (x-1)(x+h+2) \):
   \[
   (x-1)(x+h+2) = x^2 + hx + 2x - x - h - 2
   \]
   Simplifying:
   \[
   = x^2 + hx + x - h - 2
   \]

   Now, subtract the second numerator from the first:
   \[
   (x^2 + hx + (2h-1)x - 2) - (x^2 + hx + x - h - 2) = (2h-1)x - x + h = (2h-2)x + h
   \]

3. **Simplify the difference quotient expression:**
   Now, we can express \( f(x+h) - f(x) \) as:
   \[
   f(x+h) - f(x) = \frac{(2h-2)x + h}{(x+h+2)(x+2)}
   \]

4. **Divide by \( h \):**
   Finally, divide by \( h \):
   \[
   \frac{f(x+h) - f(x)}{h} = \frac{(2h-2)x + h}{h(x+h+2)(x+2)}
   \]
   Simplifying further:
   \[
   = \frac{(2h-2)x + h}{h(x+h+2)(x+2)} = \frac{(2x-2) + 1}{(x+h+2)(x+2)} = \frac{2x-1}{(x+h+2)(x+2)}
   \]

Thus, the **difference quotient** of \( f(x) = \frac{x-1}{x+2} \) is:

\[
\frac{2x-1}{(x+h+2)(x+2)}
\]

Would you like more details on any step or a further breakdown?

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Here are 5 related questions for further practice:
1. What is the derivative of \( f(x) = \frac{x-1}{x+2} \)?
2. How does the difference quotient relate to the derivative?
3. What happens to the difference quotient as \( h \to 0 \)?
4. How do you simplify difference quotients with more complex functions?
5. What is the difference quotient of \( f(x) = \frac{1}{x} \)?

**Tip**: The difference quotient is essentially the slope of the secant line between two points on the curve, and it approaches the derivative as the distance between the points shrinks (as \( h \to 0 \)).

This step-by-step guide explains how to compute the difference quotient for the rational function f(x) = (x-1)/(x+2). Learn key concepts in calculus, including the difference quotient formula, simplifying rational expressions, and how it leads to derivatives. Suitable for students in Grades 11-12.

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