Question

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

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

Answer 1

Answer: (D)

$\{0,-\sin 1\}$

$f\left(x\right)=\sin \left[x\right]$, where $\frac{-\pi }{4}<x<\frac{\pi }{4}$

We divide the range into two parts,

(a) $\frac{-\pi }{4}<x<0$

$\Rightarrow \left[x\right]=-1$

$\therefore f\left(x\right)=sin(-1)=-sin1$

(b) $0<x<\frac{\pi }{4}$

$\Rightarrow \left[x\right]=0$

$\therefore f\left(x\right)=sin0=0$

Thus, the range of $f\left(x\right)$ becomes $\{0,-sin1\}$