Question

The domain of the function $f\left(x\right)={\sin }^{-1}(x+[x\left]\right),$ where $[\cdot ]$ denotes the greatest integer function is

• A. $[0,1)$
• B. $[-1,1]$
• C. $(-1,0)$
• D. none of these

Answer 1

The correct option is A $[0,1)$

We know that the range of ${\sin }^{-1}$ is $[\frac{-\pi }{2},\frac{\pi }{2}]$ and domain is $[-1,1]$

Now in $f\left(x\right)=\sin (x+[x\left]\right)$

The input should be in the domain $[-1,1]$

$x+\left[x\right]\ge -1$

$\Rightarrow x\ge 0$

and $x+\left[x\right]\le 1$

$\Rightarrow x\le 1$

However, $\left[x\right]$ is discontinuous at $x=1$

Hence domain, of $f\left(x\right)={\sin }^{-1}(x+[x\left]\right)$ is $[0,1)$