Question

Consider $f\left(x\right)=|{\sin }^{-1}x|+{\cos }^{-1}|x|+|{\tan }^{-1}x|+{\cot }^{-1}|x|.$ If number of integers in the domain of $f$ is $n\left(d\right)$ while number of integers in the range of $f$ is $n\left(r\right)$, then $n\left(d\right)+n\left(r\right)$ is

• A. $0$
• B. $2$
• C. $3$
• D. none of these

Answer 1

The correct option is C $3$

Clearly, $D_f:[-1,1]$
Case$-1:$ $x\in [0,1]$
$f\left(x\right)={\sin }^{-1}x+{\cos }^{-1}x+{\tan }^{-1}x+{\cot }^{-1}x$
$=\frac{\pi }{2}+\frac{\pi }{2}=\pi$

Case$-2:$ $x\in [-1,0]$
$f\left(x\right)=-{\sin }^{-1}x+{\cos }^{-1}(-x)-{\tan }^{-1}x+{\cot }^{-1}(-x)$
$=\pi -{\cos }^{-1}x-{\sin }^{-1}x+\pi -{\cot }^{-1}x-{\tan }^{-1}x$
$=\pi -\frac{\pi }{2}+\pi -\frac{\pi }{2}=\pi$

$\Rightarrow R_f=\left\{\pi \right\}$
$-1,0,1\in [-1,1]$
$n\left(d\right)=3,n\left(r\right)=0$
$\Rightarrow n\left(d\right)+n\left(r\right)=3+0=3$