<Question>
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.
</Question>

<think>
<Plan_1>
Okay, so I need to find the value of f(-2) + f(-1) + f(0) where the function f(x) is given as (3x - 2)/(x - 2). Hmm, let me start by understanding the problem. The function f(x) is a rational function, meaning it's a ratio of two polynomials. The numerator is 3x - 2 and the denominator is x - 2. I need to evaluate this function at three different points: x = -2, x = -1, and x = 0, and then add those three results together. The final answer should be a common fraction.

First, let me recall how to evaluate a function at a specific value. For each x-value, I substitute it into the function's formula wherever there's an x. So, for f(-2), every x in the numerator and denominator becomes -2. Then I do the same for -1 and 0. After computing each function value, I sum them up. Seems straightforward, but let me proceed step by step to avoid mistakes.

Let me start with f(-2).

<prompt_1.1> Calculate f(-2) by substituting x = -2 into the function (3x - 2)/(x - 2). </prompt_1.1>

Next, f(-1). Substitute x = -1 into the function.

<prompt_1.2> Calculate f(-1) by substituting x = -1 into the function (3x - 2)/(x - 2). </prompt_1.2>

Then, f(0). Substitute x = 0 into the function.

<prompt_1.3> Calculate f(0) by substituting x = 0 into the function (3x - 2)/(x - 2). </prompt_1.3>
</Plan_1>

<Execution_1>
<execution_1.1>
So, f(-2) = [3*(-2) - 2]/[(-2) - 2]. Let me compute the numerator first: 3*(-2) is -6, minus 2 is -8. The denominator is (-2) - 2, which is -4. So f(-2) is (-8)/(-4). Dividing two negatives gives a positive, so that's 2.
</execution_1.1>

<execution_1.2>
So, numerator: 3*(-1) - 2 = -3 - 2 = -5. Denominator: (-1) - 2 = -3. So f(-1) = (-5)/(-3) = 5/3.
</execution_1.2>

<execution_1.3>
Let me compute each one carefully.

Starting with f(-2):

f(-2) = [3*(-2) - 2]/[(-2) - 2]

Calculating the numerator: 3*(-2) is -6, then subtract 2, so -6 - 2 = -8.

Denominator: (-2) - 2 = -4.

So f(-2) = (-8)/(-4) = 2.
</execution_1.3>
</Execution_1>

<Plan_2>
Based on execution_1.1, execution_1.2, execution_1.3:
Now f(-1):

<prompt_2.1> Sum the calculated function values f(-2), f(-1), and f(0) to find the final result. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
f(-1) = [3*(-1) - 2]/[(-1) - 2]

Numerator: 3*(-1) = -3, minus 2 is -5.

Denominator: (-1) - 2 = -3.

So f(-1) = (-5)/(-3) = 5/3.

Next, f(0):

f(0) = [3*(0) - 2]/[0 - 2]

Numerator: 0 - 2 = -2.

Denominator: 0 - 2 = -2.

So f(0) = (-2)/(-2) = 1.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait, let me verify each calculation again to make sure I didn't make any mistakes. It's easy to slip up with signs, so verification is essential.

First, f(-2):

<prompt_3.1> Verify the calculations for f(-2), f(-1), and f(0) to ensure accuracy. </prompt_3.1>

Now, adding them up: 2 + 5/3 + 1.

<prompt_3.2> Add the values of f(-2), f(-1), and f(0) to find the final result. </prompt_3.2>

Wait, but let me check if there's another way to approach this problem. Maybe simplifying the function first? Let's see.

<prompt_3.3> Explore an alternative method to verify the solution by simplifying the function (3x - 2)/(x - 2). </prompt_3.3>

But let me check once more to be thorough. Let's recompute 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) = (0 -2)/(0 -2) = (-2)/(-2) = 1. Correct.

Sum: 2 + 5/3 +1. Convert to fractions: 2 is 6/3, 1 is 3/3. So 6/3 +5/3 +3/3 = (6+5+3)/3 =14/3.

Just to make sure, maybe plug the values into the original function again:

<prompt_3.4> Re-evaluate the function values and sum to confirm the solution. </prompt_3.4>
</Plan_3>

<Execution_3>
<execution_3.1>
Numerator: 3*(-2) = -6. Then subtract 2: -6 - 2 = -8. Correct. Denominator: -2 - 2 = -4. Correct. So -8 divided by -4 is indeed 2.

f(-1): Numerator: 3*(-1) = -3. Subtract 2: -3 - 2 = -5. Denominator: -1 - 2 = -3. So -5/-3 = 5/3. Correct.

f(0): Numerator: 3*0 = 0. Subtract 2: 0 - 2 = -2. Denominator: 0 - 2 = -2. So -2/-2 = 1. Correct.
</execution_3.1>

<execution_3.2>
Let me convert everything to fractions with a common denominator. 2 is 6/3, 5/3 is already there, and 1 is 3/3. So adding them: 6/3 + 5/3 + 3/3 = (6 + 5 + 3)/3 = 14/3. So the sum is 14/3.
</execution_3.2>

<execution_3.3>
The function is f(x) = (3x - 2)/(x - 2). Let me see if I can simplify this expression. Maybe by polynomial division or factoring?

Let's try dividing 3x - 2 by x - 2.

Dividing 3x - 2 by x - 2:

First term: 3x divided by x is 3. Multiply 3 by (x - 2) gives 3x - 6. Subtract that from the numerator:

(3x - 2) - (3x - 6) = (3x - 2 - 3x + 6) = 4. So, the division gives 3 with a remainder of 4. Therefore, f(x) can be written as 3 + 4/(x - 2).

So, f(x) = 3 + 4/(x - 2). Hmm, that might be helpful. Let me verify that:

If I take 3 + 4/(x - 2), combine the terms: [3(x - 2) + 4]/(x - 2) = (3x - 6 + 4)/(x - 2) = (3x - 2)/(x - 2), which matches the original function. Great, so that's a valid simplification.

So, f(x) = 3 + 4/(x - 2). Therefore, when evaluating f at different points, this might be easier. Let's see:

f(-2) = 3 + 4/(-2 - 2) = 3 + 4/(-4) = 3 - 1 = 2. Which matches the previous result.

Similarly, f(-1) = 3 + 4/(-1 - 2) = 3 + 4/(-3) = 3 - 4/3 = (9/3 - 4/3) = 5/3.

f(0) = 3 + 4/(0 - 2) = 3 + 4/(-2) = 3 - 2 = 1. Also matches previous result. So, using the simplified form gives the same answers, which is reassuring.

Therefore, adding them up: 2 + 5/3 + 1 = 14/3.
</execution_3.3>

<execution_3.4>
For x = -2:

Numerator: 3*(-2) -2 = -6 -2 = -8.

Denominator: -2 -2 = -4.

-8 / -4 = 2. Correct.

x = -1:

Numerator: 3*(-1) -2 = -3 -2 = -5.

Denominator: -1 -2 = -3.

-5 / -3 = 5/3. Correct.

x = 0:

Numerator: 0 -2 = -2.

Denominator: 0 -2 = -2.

-2 / -2 = 1. Correct.

Adding them: 2 + 5/3 +1. Let's compute 2 +1 =3, then 3 +5/3. Since 3 is 9/3, 9/3 +5/3=14/3. So yes, 14/3.
</execution_3.4>
</Execution_3>

<Plan_4>
Based on execution_3.2, execution_3.3, execution_3.4:
Just to make sure there's no mistake in the arithmetic, let me check the addition again. 2 + 1 is 3, and 3 + 5/3 is the same as 3 + 1 2/3, which is 4 2/3, which is 14/3 as an improper fraction.

<prompt_4.1> Perform a final arithmetic check of the sum 2 + 5/3 + 1 to ensure accuracy. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
Because 4 2/3 is equivalent to (4*3 +2)/3 =14/3. Correct.

Therefore, after multiple checks and methods, I'm confident that the answer is 14/3.
</execution_4.1>
</Execution_4>

<Plan_5>
Based on execution_4.1:
**Final Answer**
The value of \(f(-2) + f(-1) + f(0)\) is \boxed{\dfrac{14}{3}}.
</Plan_5>

<Execution_5>
No prompt to execute.
</Execution_5>

<Final_answer>
Therefore, the value of $f(-2) + f(-1) + f(0)$, expressed as a common fraction, is $\boxed{\frac{14}{3}}$.
</Final_answer>
</think>