FunctionHard
Question
The sum of all the elements in the range of $f(x) = Sgn(sinx) + Sgn(cosx) + Sgn(tanx) + Sgn(cotx)$, $x \neq \frac{n\pi}{2},n \in \mathbf{Z}$,
where $Sgn(t) = \left\{ \begin{matrix} 1, & \text{~}\text{if}\text{~} & t > 0 \\ - 1 & \text{~}\text{if}\text{~} & t < 0 \end{matrix} \right.\ $, is
Options
A.4
B.2
C.-2
D.0
Solution
$x \in (0,\pi/2) \Rightarrow y = 1 + 1 + 1 + 1 = 4$
$${x \in (\pi/2,\pi) \Rightarrow y = 1 - 1 - 1 - 1 = - 2 }{x \in (\pi,3\pi/2) \Rightarrow y = - 1 - 1 + 1 + 1 = 0 }{x \in (3\pi/2,2\pi) \Rightarrow y = - 1 + 1 - 1 - 1 = - 2 }$$∴ Range of $y$ is $\{ - 2,0,4\}$
Required sum $= - 2 + 0 + 4 = 2$
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