Question

The range of $f\left(x\right)=\cos \left[x\right],-\frac{\pi }{4}<x<\frac{\pi }{4}$,where $\left[\right]$ represent greatest integer function less than or equal to $x$ is

• A. $\left\{0\right\}$
• B. $[-1,1]$
• C. $\{\cos 1,1\}$
• D. $\{-1,1\}$

Answer 1

The correct option is C $\{\cos 1,1\}$

Given $f\left(x\right)=\cos \left[x\right]$
Also given $-\frac{\pi }{4}<x<\frac{\pi }{4}$
$-\frac{\pi }{4}\le x<0\cup 0\le x<\frac{\pi }{4}$
Now, $xϵ\left[\frac{-\pi }{4},0\right)$ i.e. $\frac{-\pi }{4}\le x<0$
$\cos \left[x\right]=\cos (-1)=\cos 1$
and $0\le x\le \frac{\pi }{4},$ we have
$\cos \left[x\right]=\cos \left(0\right)=1$
Required range$=\{\cos 1,1\}$