## Text Analysis: Annotated Reasoning Chain
### Overview
The image presents an annotated reasoning chain for solving a mathematical problem. It shows the step-by-step solution to finding the value of f(-2) + f(-1) + f(0) given the function f(x) = (3x-2)/(x-2). The solution is presented as a dialogue within a "<think>" block, demonstrating the reasoning process.
### Components/Axes
* **Prompt:** The initial problem statement.
* **Reasoning Chain:** A step-by-step explanation of the solution process.
* **Final Answer:** The boxed final answer to the problem.
### Detailed Analysis or ### Content Details
**Prompt:**
"If $f(x) = \frac{3x-2}{x-2}$, what is the value of $f(-2) +f(-1)+f(0)$? Express your answer as a common fraction. Please reason step by step, and put your final answer within \boxed{x}."
**Reasoning Chain:**
The reasoning chain is presented within a `<think>` block. It can be broken down into the following steps:
1. **Understanding the Problem:** The problem is identified as finding the value of f(-2) + f(-1) + f(0) where f(x) = (3x-2)/(x-2).
2. **Finding f(-2):**
* Substitute x = -2 into the function: \[3*(-2) - 2]/[(-2) - 2].
* Numerator: 3*(-2) = -6, then -6 - 2 = -8.
* Denominator: -2 - 2 = -4.
* f(-2) = -8 / -4 = 2.
3. **Finding f(-1):**
* Substitute x = -1 into the function.
* Numerator: 3*(-1) - 2. 3*(-1) = -3, then -3 - 2 = -5.
* Denominator: (-1) - 2 = -3.
* f(-1) = -5 / -3 = 5/3.
4. **Finding f(0):**
* Substitute x = 0 into the function.
* Numerator: 3*0 - 2 = 0 - 2 = -2.
* Denominator: 0 - 2 = -2.
* f(0) = -2 / -2 = 1.
5. **Adding the Values:**
* f(-2) + f(-1) + f(0) = 2 + 5/3 + 1.
* Convert to thirds: 2 = 6/3, 1 = 3/3.
* 6/3 + 5/3 + 3/3 = (6 + 5 + 3)/3 = 14/3.
6. **Verification:**
* Double-checking each function value:
* f(-2): (3*(-2) - 2)/(-2 - 2) = (-6 - 2)/(-4) = (-8)/(-4) = 2. Correct.
* f(-1): (3*(-1) - 2)/(-1 - 2) = (-3 - 2)/(-3) = (-5)/(-3) = 5/3. Correct.
* f(0): (3*0 - 2)/(0 - 2) = (0 - 2)/(-2) = (-2)/(-2) = 1. Correct.
* Adding them up again: 2 + 5/3 + 1 = 14/3.
**Final Answer:**
"The value of (f(-2) + f(-1) + f(0) \) is \(\boxed\dfrac{14}{3}\)."
### Key Observations
* The solution is presented in a conversational style, simulating a thought process.
* Each step is clearly explained, including the substitution and simplification of the function.
* The solution includes a verification step to ensure accuracy.
### Interpretation
The annotated reasoning chain demonstrates a clear and methodical approach to solving the given mathematical problem. The step-by-step explanation makes the solution easy to follow and understand. The inclusion of verification steps highlights the importance of accuracy and attention to detail. The conversational style makes the solution more accessible and engaging.
