Step 1: Understanding the Topic and Question
The provided equation, $f(x) = ( f(x) - \pi x ) + \pi$, simplifies algebraically to $\pi x = \pi$, or $x=1$. This indicates that the equation as written is not a functional equation valid for all $x$, but rather a condition that holds only at $x=1$. The question is likely ill-posed or contains a typographical error. For instance, it might involve a non-standard function like the floor or ceiling function, e.g., $f(x) = \lfloor f(x) - \pi x \rfloor + \pi$. The provided solution implies such a periodic or discontinuous nature.
Step 2: Key Approach - Interpreting the Problem's Intent
Given the structure of the answers, we will proceed by interpreting the problem's likely intent, as reflected in the provided solution. The solution suggests a relationship where the difference $f(x+1) - f(x)$ is not constant but has some ambiguity or multi-valued nature. The term $\pi$ suggests a base difference, and the $\pm 1/6$ suggests a periodic adjustment.
Step 3: Detailed Explanation (Following the Solution's Logic)
The problem is interpreted to describe a function with a specific periodic behavior. We need to find the value of $f(3) - f(2)$, which is the change in the function over a unit interval.
The structure of the flawed equation and the answers suggest that the function $f(x)$ increases by approximately $\pi$ for every unit increase in $x$.
The solution implies that this increase is not exact but subject to a periodic fluctuation, leading to two possible outcomes. This kind of behavior is characteristic of functions involving step-functions or branch cuts.
The solution concludes that this ambiguity leads to a difference of $\pi$ plus or minus a small value, given as $1/6$ in the options.
\[
f(3) - f(2) = \pi \pm \frac{1}{6}
\]
Step 4: Final Answer
Based on this interpretation of the problem's implicit properties, the possible values for the difference $f(3) - f(2)$ are those given in options (A) and (B).
\[
\pi + \frac{1}{6} \quad \text{and} \quad \pi - \frac{1}{6}
\]