Question

Find the domain and range of the function given by $f\left(x\right)=\frac{1}{\sqrt{x-\left[x\right]}}$, where [x] is greatest integer function.

Answer 1

We have, $f\left(x\right)=\frac{1}{\sqrt{x-\left[x\right]}}$

for domain of f:we know that,

$0\le x-\left[x\right],1,\forall xϵR$

Also, $x-\left[x\right]=0\forall xϵZ$

$\therefore 0<x-\left[x\right]<1,\forall xϵR-Z$

$\Rightarrow f\left(x\right)=\frac{1}{\sqrt{x-\left[x\right]}}\forall xϵR-Z$

$\therefore \text{Domain of}f\left(x\right)=R-Z$

For range of f:we have $0<x-\left[x\right]<1,\forall xϵR-Z$

$\Rightarrow 0<\sqrt{x-\left[x\right]}<1,\forall xϵR-Z$

$\Rightarrow 1<\frac{1}{\sqrt{x-\left[x\right]}}<\infty ,\forall xϵR-Z$

$\Rightarrow 1<f\left(x\right)<\infty ,\forall xϵR-Z$

$\therefore \text{Range of}f\left(x\right)=(1,\infty )$