Question

$f\left(x\right)$ is a real valued function
$\&f\left(x\right)=\left[{\log }_{\sqrt{2}}(\sin \left(\ln \left(\frac{\sqrt{4-x^2}}{1-x}\right)\right)+\cos \left(\ln \left(\frac{\sqrt{4-x^2}}{1-x}\right)\right))\right]$
where [.] denotes the greatest integer function.
Then which of the following statements are true?

• A. Number of integers in the domain of $f\left(x\right)$ is 2.
  
• B. Number of integers in the domain of $f\left(x\right)$ is 3.
  
• C. Range of $f\left(x\right)$ is set of all non positive integers $\cup$ {1}
  
• D. Range of $f\left(x\right)$ is set of all positive integers

Answer 1

Answer: (A), (C)

The correct options are
A Number of integers in the domain of $f\left(x\right)$ is 2.

C Range of $f\left(x\right)$ is set of all non positive integers $\cup$ {1}


$\begin{array}{cc}4-x^2>0,& 1-x>0\\ -2<x<2,& n<1\end{array}$

So domain is -2<x<1, so integers are -1, 0

For range,

$0<\frac{\sqrt{4-x^2}}{1-x}<\infty$

$-\infty <ln\left(\frac{\sqrt{4-x^2}}{1-x}\right)<\infty$

$\Rightarrow -\sqrt{2}\le \sin \left(\ln \frac{\sqrt{4-x^2}}{1-x}\right)+\cos \left(ln\left(\frac{\sqrt{4-x^2}}{1-x}\right)\right)\le \sqrt{2}$

So $lo{g}_{\sqrt{2}}\left(\frac{ln\sqrt{4-x^2}}{1-x}\right)ϵ[-\infty ,1]$

So Range of $\left[lo{g}_{\sqrt{2}}\left(\frac{\ln \sqrt{4-x^2}}{1-x}\right)\right]$ is set of all non positive integers $\cup${1}