Question

For the function $f\left(x\right)=sin\left(\pi \left[x\right]\right)\times co{s}^{-1}\left(\left[x\right]\right)$ , choose the correct option.

(where $\left[.\right]$ represents the greatest integer function)

• A. Domain of $f\left(x\right)\in \left[-1,1\right]$
• B. Range of $f\left(x\right)$ contains exactly two elements
• C. $f\left(x\right)$ is an identity function
• D. $f\left(x\right)$ is a constant function

Answer 1

Answer: (D)

For any $x\in R,$ we know,
$\sin \pi [x]=0$
$\therefore f(x)=0,\forall x\in [-1,2)\{\because -1\le [x]\le 1$ for ${\cos }^{-1}[x]$ to be defin ed $\}$
$\therefore$ Domain $x\in [-1,2)$
$\Rightarrow$ Range $=\{0\}$
$\therefore f(x)$ is a constant function