Question

Let $f(x):cosx(cosx+sinx)=[x]$, where $[\cdot ]$ represent the greatest integer function, the interval for which $x$ is domain of $f(x)$, is

• A. $[0,1)\cup [1,2]$
• B. $0lex<1\cup 1\le x<2$
• C. $0<x<1\cup 1<x<2$
• D. $-1<x<1\cup 1\le x<2$

Answer 1

Answer: (B)

$f(x):\cos x(\cos x+\sin x)=[x]$
$\Rightarrow {\cos }^{2}x+\sin x\cos x=[x]$
$\Rightarrow \cos 2x+\sin 2x=2[x]-1$ (Multiply by 2 on both sides
and use formula $\cos 2x=2{\cos }^{2}x-1)$
$\Rightarrow \sqrt{2}\sin \left(2x+\frac{\pi }{4}\right)=2[x]-1$
$\Rightarrow -\sqrt{2}\le 2[x]-1\le \sqrt{2}$
$\Rightarrow \frac{1-\sqrt{2}}{2}\le [x]\le \frac{\sqrt{2}+1}{2}$
$\Rightarrow [x]=0,1\therefore [x]=0,[x]=1$
$\Rightarrow x\in [0,1)$ and $x\in [1,2)$