Functions Part 2

Concept:The sign function $\operatorname{Sgn}(t)$ gives $1$ if $t>0$ and $-1$ if $t<0$.The signs of $\sin x$, $\cos x$, $\tan x$, and $\cot x$ depend only on the quadrant of $x$.We evaluate $f(x)$ quadrant by quadrant, then find all distinct values and sum them.Explanation:We consider four intervals (quadrants) for $x$, excluding points where any function is zero ($x \neq \frac{n\pi}{2}$).In each quadrant, we determine the sign of each trigonometric function and then add the corresponding $\operatorname{Sgn}$ values.From the table, the possible values of $f(x)$ are $4$, $-2$, and $0$.Thus, the range of $f(x)$ is $\{-2, 0, 4\}$.Sum of all elements in the range = $4 + (-2) + 0 = 2$.Answer:$2$ (Option C)