Mathematics - Functions Question with Solution | TestHub

The function is defined if (i) $[\mathrm{x}]>0$ and $[\mathrm{x}]\ne 1$

(ii) $\frac{|x|}{x}>0$

(iii) ${\log }_{[x]}\frac{|x|}{x}\ge 0$ (iv) $0\le \sqrt{{\log }_{[x]}\frac{|x|}{x}}\le 1$.

Now, (i) $[\mathrm{x}]=2,3,\dots$ i.e. $[x]\ge 2\Rightarrow x\ge 2$ i.e. the domain of the function is $[2,\infty )$. For this value of $\mathrm{x}(\ge 2)$ (ii) is true, (iii) is also true and $\sqrt{{\log }_{[x]}\frac{|x|}{x}}=\sqrt{{\log }_{[x]}1}=0$.

Hence $f(x)={\cos }^{-1}(0)=\frac{\pi }{2}$.

Hence the range of the function is $\left\{\frac{\pi }{2}\right\}$.