Question

Find the domain of the following function:

$f\left(x\right)=cose{c}^{-1}[1+{\sin }^{2}x],$ where $[\cdot ]$ denotes the greatest integer function

Answer 1

Given function is, $f\left(x\right)=cose{c}^{-1}[1+{\sin }^{2}x]$
For this function to be defined,

$\left|\right[1+{\sin }^{2}x\left]\right|\ge 1$
$\Rightarrow [1+{\sin }^{2}x]\ge 1,$ (Since $[1+{\sin }^{2}x]>0$)
$\Rightarrow 1+\left[{\sin }^{2}x\right]\ge 1$

$\Rightarrow \left[{\sin }^{2}x\right]\ge 0$
Clearly this inequality is true for all real number

Hence, the domain is $xϵR$