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  4. Find the domain of the real valued function f(x)=([x2]-[x]-2)-1 / 2, where [.] is the greatest integer function

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Q. Find the domain of the real valued function $f(x)=\left(\left[x^{2}\right]-[x]-2\right)^{-1 / 2}$, where $[.]$ is the greatest integer function

AP EAMCETAP EAMCET 2020

Solution:

Let $y=f(x)=\left([x]^{2}-[x]-2\right)^{-1 / 2}$
$\Rightarrow y^{2}=\frac{1}{\sqrt{[x]^{2}-[x]-2}}$
For real valued
$[x]^{2}-[x]-2>0$
$\Rightarrow \{[x]-2\}\{[x]+1\} >0$
$[x] \in R-(-1,2)$
So, $x \in R-(-1,3]$