# Technical Document Extraction and Analysis
## Problem Statement
Given the function $ f(x) = \frac{3x - 2}{x - 2} $, compute the value of $ f(-2) + f(-1) + f(0) $ and express the result as a common fraction.
---
## Attempt 1 Analysis
### Steps and Calculations
1. **Evaluate $ f(-2) $:**
$$
f(-2) = \frac{3(-2) - 2}{-2 - 2} = \frac{-6 - 2}{-4} = \frac{-8}{-4} = 2
$$
- **Error Identified:** The attempt incorrectly states $ \frac{-8}{-4} = -1 $, which is mathematically invalid. The correct result is $ 2 $.
2. **Evaluate $ f(-1) $:**
$$
f(-1) = \frac{3(-1) - 2}{-1 - 2} = \frac{-3 - 2}{-3} = \frac{-5}{-3} = \frac{5}{3}
$$
- **Correct Calculation.**
3. **Evaluate $ f(0) $:**
$$
f(0) = \frac{3(0) - 2}{0 - 2} = \frac{-2}{-2} = 1
$$
- **Correct Calculation.**
4. **Summation:**
$$
f(-2) + f(-1) + f(0) = 2 + \frac{5}{3} + 1 = \frac{6}{3} + \frac{5}{3} + \frac{3}{3} = \frac{14}{3}
$$
- **Final Answer:** $ \boxed{\frac{14}{3}} $ (Correct, but the attempt incorrectly claims the answer is $ \boxed{\frac{8}{3}} $).
---
## Attempt 2 Analysis
### Steps and Calculations
1. **Evaluate $ f(-2) $:**
$$
f(-2) = \frac{3(-2) - 2}{-2 - 2} = \frac{-8}{-4} = 2
$$
- **Correct Calculation.**
2. **Evaluate $ f(-1) $:**
$$
f(-1) = \frac{3(-1) - 2}{-1 - 2} = \frac{-5}{-3} = \frac{5}{3}
$$
- **Correct Calculation.**
3. **Evaluate $ f(0) $:**
$$
f(0) = \frac{3(0) - 2}{0 - 2} = \frac{-2}{-2} = 1
$$
- **Correct Calculation.**
4. **Summation:**
$$
f(-2) + f(-1) + f(0) = 2 + \frac{5}{3} + 1 = \frac{14}{3}
$$
- **Final Answer:** $ \boxed{\frac{14}{3}} $ (Correct, but the attempt incorrectly claims the answer is $ \boxed{5} $).
---
## Attempt 3 Analysis
### Steps and Calculations
1. **Evaluate $ f(-2) $:**
$$
f(-2) = \frac{3(-2) - 2}{-2 - 2} = \frac{-8}{-4} = 2
$$
- **Correct Calculation.**
2. **Evaluate $ f(-1) $:**
$$
f(-1) = \frac{3(-1) - 2}{-1 - 2} = \frac{-5}{-3} = \frac{5}{3}
$$
- **Correct Calculation.**
3. **Evaluate $ f(0) $:**
$$
f(0) = \frac{3(0) - 2}{0 - 2} = \frac{-2}{-2} = 1
$$
- **Correct Calculation.**
4. **Summation:**
$$
f(-2) + f(-1) + f(0) = 2 + \frac{5}{3} + 1 = \frac{14}{3}
$$
- **Final Answer:** $ \boxed{\frac{14}{3}} $ (Correct).
---
## Conclusion
- **Correct Answer:** $ \boxed{\frac{14}{3}} $.
- **Errors in Attempts 1 and 2:**
- Attempt 1 miscalculated $ f(-2) $ as $ -1 $ instead of $ 2 $.
- Attempt 2 incorrectly summed $ 2 + \frac{5}{3} + 1 $ as $ 5 $ instead of $ \frac{14}{3} $.
- **Attempt 3** is fully correct in both calculations and summation.
