Question

Find the domain of the following function:

$f\left(x\right)={\cos }^{-1}\sqrt{\log \left[x\right]\frac{\left|x\right|}{x}},$ where,$[\cdot ]$ denotes the greatest integer function.

Answer 1

Given, $f\left(x\right)={\cos }^{-1}\sqrt{\log \left[x\right]\frac{\left|x\right|}{x}}$
For given function to be defined $\left[x\right]>0\Rightarrow x\ge 1$
$\therefore \frac{|x|}{x}=1$
$\Rightarrow f\left(x\right)={\cos }^{-1}\sqrt{\log \left[x\right]}$
Also for square root to be defined $\log \left[x\right]\ge 0\Rightarrow \left[x\right]\ge 1\Rightarrow x\ge 1$
And for ${\cos }^{-1}$ to be defined $\sqrt{\log \left[x\right]}\le 1\Rightarrow \log \left[x\right]\le 1\Rightarrow \left[x\right]<e\Rightarrow x<3$
Hence domain of $f\left(x\right)$ is $[1,3)$