Question

$f:R\to [-1,\infty )$ and $f\left(x\right)=ln\left(\right[|\sin 2x|+|\cos 2x|\left]\right)$ (where [.] denotes greatest integer function) then which of the following is NOT correct?

• A. $(R^{-}\cap Rangeoff(x\left)\right)$ is a null set
• B. f(x) is periodic but fundamental period is not defined.
• C. f(x) is invertible in $[0,\frac{\pi }{4}]$
• D. f(x) is into function.

Answer 1

The correct option is C f(x) is invertible in $[0,\frac{\pi }{4}]$

$\because 1\le |\sin 2x|+|\cos 2x|\le \sqrt{2}So,[|sin2x|+|cos2x|]=1f\left(x\right)=ln\left(1\right)=0$
$\Rightarrow$ it is periodic but fundamental period is not defined.
Also, $(R^{-}\cap Rangeoff(x\left)\right)$ is a null set
f(x) is many one and into fucntion.
Hence, f(x) is not invertible.