Question

Let $f(x)=\operatorname{sgn}(\operatorname{arccot} x)+\tan \left(\frac{\pi}{2}[x]\right)$, where $[x]$ is the greatest integer function less than or equal to $x$. Then which of the following alternatives is/are true?

• A. $ f(x)$ is many one but not even function
• B. $f(x)$ is periodic function
• C. $f ( x )$ is bounded function
• D. Graph of $f(x)$ remains above $x$-axis

Answer 1

Answer: (A), (B), (C), (D)

Clearly domain of $f(x)=\underset{n \in I}{Y}[2 n, 2 n+1)$
$\Rightarrow f ( x )=1, \forall x \in D _{ f }$

Graph of $f(x)=\operatorname{sgn}\left(\cot ^{-1} x\right)+\tan \frac{\pi}{2}[x]$
From graph $f ( x )$ is periodic with period 2.
$\Rightarrow $ Option(s) (A), (B), (C), (D) are correct