Question

The domain of $f\left(x\right)={\log }_5(x-[x\left]\right)$ is
(where [.] denotes the greatest integer function)

• A. $R-Z$
• B. $R$
• C. $R^{+}-W$
• D. $(0,\infty )$

Answer 1

The correct option is A $R-Z$

Let $f\left(x\right)={\log }_5(x-[x\left]\right)$
For $\log$ to be defined
$(x-[x\left]\right)>0$

We know that, $\left[x\right]=x$ if $x\in Z$
and $x\ge \left[x\right]\forall x\in R$
$\Rightarrow x-\left[x\right]\ge 0\forall x\in R$
​​​​​​​$\Rightarrow x-\left[x\right]>0\forall x\in R-Z$