Question

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

• A. $R$
• B. $R-Z$ where $Z$ is the set of all integers
• C. $(0,+\infty )$
• D. None of these

Answer 1

The correct option is B $R-Z$ where $Z$ is the set of all integers

The log input must not be 0 or negative.That is
$x-\left[x\right]>0$
$x-\left[x\right]=0$ for all $xϵZ$ ...(where Z is the set of integers).
And $x-\left[x\right]>0$ for $xϵR-Z$
Hence the required domain for $f\left(x\right)=lo{g}_e(x-[x\left]\right)$
is $R-Z$.