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)

• A. $[-1,1]$
• B. $R$
• C. $\left\{1\right\}$
• D. $R-\left\{1\right\}$

Answer 1

The correct option is C $\left\{1\right\}$

$f\left(x\right)=\frac{[x-3]}{\left[x\right]-3}$
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, $R\left(f\right)=\left\{1\right\}$