1. The problem involves understanding the function $f: S \to S$ where $S = \{1, 2, 3, \ldots, 12\}$ and $f$ is onto (surjective). We are asked to analyze or solve something related to $x - 3x - f(x)$ and the function $f$. 2. Since $f$ is onto from $S$ to $S$, every element in $S$ has at least one preimage in $S$. This means $f$ is a surjection on a finite set of size 12. 3. The expression $x - 3x - f(x)$ simplifies algebraically to $-2x - f(x)$. This might be part of a larger problem or equation involving $f$. 4. Without additional equations or conditions, we can only state that for each $x \in S$, the value $-2x - f(x)$ is defined, and $f(x)$ takes values in $S$. 5. If the problem is to find $f(x)$ or analyze the function further, more information is needed. However, given the surjectivity and the set $S$, $f$ is a permutation or a surjective mapping on $S$. Final answer: The expression simplifies to $$-2x - f(x)$$ where $f$ is an onto function from $S$ to $S$ with $S = \{1,2,\ldots,12\}$.