Question

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

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

Answer 1

The correct option is B $\left\{1,\cos 1\right\}$

When $-\frac{\pi }{4}<x<0\Rightarrow -1<-\frac{\pi }{4}<x<0$

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

$\Rightarrow f\left(x\right)=\cos [-1]=\cos 1$

When $0\le x\le \frac{\pi }{4}<1$

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

$\Rightarrow f\left(x\right)=\cos 0=1$

$\therefore R_f=1\cos 1=\cos 1$

Option$\left(b\right)$ is correct.