# Technical Document Extraction: Mathematical Problem and Solution Attempts This document contains a transcription and analysis of an image depicting a mathematical problem followed by three distinct "Attempts" at a solution. The text uses LaTeX formatting for mathematical expressions. ## 1. Problem Statement **Text:** 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. --- ## 2. Component Isolation: Attempt 1 This section contains a step-by-step calculation with a final answer highlighted in red. **Transcription:** * Let's evaluate $f(-2)$. * We have $f(-2) = \frac{3(-2)-2}{-2-2} = -\frac{4}{-4} = \frac{1}{1} = 1$. * So $f(-2) = 1$. * Let's evaluate $f(-1)$. * We have $f(-1) = \frac{3(-1)-2}{-1-2} = -\frac{5}{-3} = \frac{5}{3}$. * So $f(-1) = \frac{5}{3}$. * Let's evaluate $f(0)$. * We have $f(0) = \frac{3(0)-2}{0-2} = \frac{-2}{-2} = 1$. * So $f(0) = 1$. * So we want to compute $1 + \frac{5}{3} + 1 = \frac{3}{3} + \frac{5}{3} + \frac{3}{3} = \frac{3+5+3}{3} = \frac{11}{3}$. (Note: Transcription reflects the logic provided in the image text). * So the answer is $\boxed{\frac{8}{3}}$. * #### $\frac{8}{3}$ **Technical Note:** This attempt contains an arithmetic error in the first step: $3(-2)-2$ is calculated as $-4$ instead of $-8$. --- ## 3. Component Isolation: Attempt 2 This section contains an alternative calculation with a final answer highlighted in red. **Transcription:** * Let's first find $f(-2)$. * $f(-2) = \frac{3(-2)-2}{-2-2} = \frac{-6-2}{-4} = \frac{-8}{-4} = 2$. * Now let's find $f(-1)$. * $f(-1) = \frac{3(-1)-2}{-1-2} = \frac{-3-2}{-3} = \frac{-5}{-3} = \frac{5}{3}$. * Now let's find $f(0)$. * $f(0) = \frac{3(0)-2}{0-2} = \frac{-2}{-2} = 1$. * So we have $f(-2) + f(-1) + f(0) = 2 + \frac{5}{3} + 1 = \frac{7}{3} + \frac{5}{3} + \frac{3}{3} = \frac{7+5+3}{3} = \frac{15}{3} = 5$. * So the answer is $\boxed{5}$. * #### 5 **Technical Note:** This attempt contains an arithmetic error in the summation step: $2 + 1$ is incorrectly converted to $\frac{7}{3} + \frac{3}{3}$ (where $2$ should be $\frac{6}{3}$). --- ## 4. Component Isolation: Attempt 3 This section contains a third calculation with a final answer highlighted in red. **Transcription:** * Let's plug in $-2$, $-1$, and $0$ for $x$ in $f(x)$. * We get $f(-2) = \frac{3(-2)-2}{-2-2} = \frac{-8}{-4} = \frac{2}{1} = 2$, $f(-1) = \frac{3(-1)-2}{-1-2} = \frac{-5}{-3} = \frac{5}{3}$, $f(0) = \frac{3(0)-2}{0-2} = \frac{-2}{-2} = 1$. * So we have $2 + \frac{5}{3} + 1 = \frac{6}{3} + \frac{5}{3} + \frac{3}{3} = \frac{14}{3}$. * So the answer is $\boxed{\frac{14}{3}}$. * #### $\frac{14}{3}$ **Technical Note:** This attempt appears to be mathematically consistent. $2 + \frac{5}{3} + 1 = 3 + \frac{5}{3} = \frac{9}{3} + \frac{5}{3} = \frac{14}{3}$. --- ## 5. Summary of Data | Attempt | $f(-2)$ | $f(-1)$ | $f(0)$ | Final Sum Result | | :--- | :--- | :--- | :--- | :--- | | **Attempt 1** | 1 (Error) | $5/3$ | 1 | $8/3$ | | **Attempt 2** | 2 | $5/3$ | 1 | 5 (Error) | | **Attempt 3** | 2 | $5/3$ | 1 | $14/3$ |