Question #40372

This question tests the 'Domain Nuance' and 'Ghost Range' trap associated with the Signum function.

Students often assume $\text{sgn}(g(x))$ will output $0$ at the roots of $g(x)$.

However, if the roots of $g(x)$ are explicitly excluded from the function's domain (due to the rational function's denominator), the value $0$ is never achieved.

Here, simplifying the rational expression yields $x-2$, tempting students to evaluate the function at the excluded points $x=1$ and $x=2$.

Doing so produces the 'ghost' range values $-2$ and $0$, leading directly to the primary distractor.

Rigorous domain establishment ($D = \mathbb{R} \setminus \{1, 2\}$) is required before applying the signum operator.