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 step by step. First, I need to compute each of these three function values separately and then add them up. That sounds straightforward, but I should make sure I don't make any arithmetic mistakes. Let me take it one at a time.

Starting with f(-2). Let me substitute x = -2 into the function. So, the numerator becomes 3*(-2) - 2, and the denominator is (-2) - 2. Let me compute the numerator first: 3 times -2 is -6, and then subtract 2, so that's -6 - 2 = -8. The denominator is -2 - 2, which is -4. Therefore, f(-2) is -8 divided by -4. Dividing two negatives gives a positive, so that's 2. Okay, so f(-2) is 2. That seems right.

Next, f(-1). Substitute x = -1 into the function. Numerator: 3*(-1) - 2. That's -3 - 2 = -5. Denominator: -1 - 2 = -3. So f(-1) is -5 divided by -3. Again, both negatives, so that's 5/3. Wait, but 5 divided by 3 is 1 and 2/3, but since it's negative over negative, it's positive. So f(-1) is 5/3. Let me confirm that: 3*(-1) is -3, minus 2 is -5. Denominator: -1 -2 is -3. So yes, -5/-3 = 5/3. Correct.

Now f(0). Substitute x = 0. Numerator: 3*0 - 2 = 0 - 2 = -2. Denominator: 0 - 2 = -2. So f(0) is -2/-2. That's 1. Because -2 divided by -2 is 1. That makes sense.

So adding them up: f(-2) + f(-1) + f(0) = 2 + 5/3 + 1. Let me compute that. First, 2 and 1 are whole numbers. 2 + 1 = 3. Then add 5/3. So 3 is equal to 9/3. 9/3 + 5/3 = 14/3. Wait, 3 + 5/3. Let me check again. 2 + 5/3 + 1. So 2 + 1 is 3, which is 9/3. Then 9/3 + 5/3 is 14/3. So the total is 14/3. Hmm. Let me verify each step again to make sure I didn't make a mistake.

First, f(-2): numerator 3*(-2) -2 = -6 -2 = -8. Denominator -2 -2 = -4. So -8/-4 = 2. Correct. f(-1): numerator 3*(-1) -2 = -3 -2 = -5. Denominator -1 -2 = -3. So -5/-3 = 5/3. Correct. f(0): numerator 0 -2 = -2. Denominator 0 -2 = -2. So -2/-2 = 1. Correct. Then adding 2 + 5/3 + 1. 2 +1 is 3, which is 9/3. 9/3 +5/3 =14/3. So 14/3 is the answer. That seems right. Let me check once more.

Alternatively, maybe I can compute each term as fractions to ensure there's no error. Let's see:

f(-2) = (-8)/(-4) = 2/1 = 2. So 2 is 6/3. Wait, 2 is 6/3. Then f(-1) is 5/3. f(0) is 1, which is 3/3. So adding them up: 6/3 + 5/3 + 3/3 = (6 +5 +3)/3 =14/3. Yep, same result. So that's consistent.

Alternatively, maybe if I convert all to improper fractions: 2 is 2/1, which is 6/3. 5/3 is already a fraction, and 1 is 3/3. So adding 6/3 +5/3 +3/3 =14/3. Correct. So the answer should be 14/3. Let me make sure there's no calculation mistake in the function evaluations.

Wait, another way to check: Let's redo each function evaluation.

For f(-2):

(3*(-2) -2)/(-2 -2) = (-6 -2)/(-4) = (-8)/(-4) = 2. Correct.

For f(-1):

(3*(-1) -2)/(-1 -2) = (-3 -2)/(-3) = (-5)/(-3) = 5/3. Correct.

For f(0):

(3*0 -2)/(0 -2) = (-2)/(-2) = 1. Correct.

Yes, that's all correct. So adding them up: 2 + 5/3 +1. Converting to fractions with denominator 3: 2 is 6/3, 1 is 3/3. So 6/3 +5/3 +3/3 =14/3. So 14/3 is the final answer. That makes sense. I think that's correct. I don't see any errors in the calculations. So the value of f(-2) +f(-1)+f(0) is 14/3.
Therefore, the value of $f(-2) + f(-1) + f(0)$ is $\boxed{\frac{14}{3}}$.
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