Question

Let $f(x)=\mathrm{sgn}(\mathrm{arccot}x)+\tan \left(\frac{\pi }{2}[x]\right)$, 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 $f(x)=\underset{n\in I}{Y}[2n,2n+1)$
$\Rightarrow f(x)=1,\forall x\in D_f$

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