Question

The range of the function $f\left(x\right)=\frac{[x-3]}{\left[x\right]-3}$ in its domain is
($[.]$ represents the greatest integer function)

Answer 1

$f\left(x\right)=\frac{[x-3]}{\left[x\right]-3}$

For $f$ to be defined,
$\left[x\right]-3\ne 0$
$\Rightarrow x\notin [3,4)$

Hence, domain of $f$ is $R-[3,4)$

$f\left(x\right)=\frac{[x-3]}{\left[x\right]-3}$
$\Rightarrow f\left(x\right)=\frac{\left[x\right]-3}{\left[x\right]-3}(\because [x+m]=[x]+m\forall m\in Z)$
$\Rightarrow f\left(x\right)=1$ for all $x$ in the domain of $f$.

Hence, $R\left(f\right)=\left\{1\right\}$