Question

If $f\left(x\right)=2\left[x\right]+\cos ([-\pi ]x)$, where $[.]$ represents greatest integer function, then

• A. $f\left(\frac{\pi }{2}\right)=3$
• B. $f\left(\pi \right)=0$
• C. $f\left(\frac{\pi }{4}\right)=-1$
• D. $f(-\pi )=-7$

Answer 1

Answer: (A), (C), (D)

$f(-\pi )=-7$

$f\left(x\right)=2\left[x\right]+\cos ([-\pi ]x)$
$\Rightarrow f\left(x\right)=2\left[x\right]+\cos (-4x)\Rightarrow f\left(x\right)=2\left[x\right]+\cos \left(4x\right)$

Now,
$f\left(\frac{\pi }{2}\right)=2\left[\frac{\pi }{2}\right]+\cos \left(2\pi \right)=2\cdot 1+1=3$

$f\left(\pi \right)=2\left[\pi \right]+\cos \left(4\pi \right)=2\cdot 3+1=7$

$f\left(\frac{\pi }{4}\right)=2\left[\frac{\pi }{4}\right]+\cos \left(\pi \right)=2\cdot 0-1=-1$

$f(-\pi )=2[-\pi ]+\cos (-4\pi )=2(-4)+1=-7$