Question

The range of the function $f(x)= \sin [x],-\frac {\pi}{4} < x < \frac {\pi}{4}$ where $[x]$ denotes the greatest integer $\leq x$, is ________

• A. $\{0\}$
• B. $\{0,-1\}$
• C. $\{0, \pm sin\, 1\}$
• D. $\{0, - sin 1\}$

Answer 1

Answer: (D)

Given, $f(x)=\sin [x],-\frac{\pi}{4}< x< \frac{\pi}{4}$
Clearly, $\sin 0=0$
and $\left[\frac{\pi}{4}\right]=\left[\frac{3.14}{4}\right]=0 $
$\therefore \forall x \in\left[0, \frac{\pi}{4}\right], \sin [x]=0$
$\forall x \in\left[-\frac{\pi}{4}, 0\right),[x]=-1 $
$ \therefore \sin [x]=\sin (-1)=-\sin 1$
So, the range of $f(x)$ is $\{0,-\sin 1\}$.