Question

The domain of the function $f(x)=\sqrt{x^2−[x{]}^2}$, where $[x]$ denotes the greatest integer less than or equal to $x$, is

• A. $[0,∞)$
• B. $(−∞,0)$
• C. $(−∞,∞)$
• D. None of these

Answer 1

Answer: (A)

If $xâ‰\yen 0$, then $[x]$ is an integer part of $x$
Hence $x^2â‰\yen [x{]}^2$ and $f(x)=\sqrt{x^2−[x{]}^2}$ is well-defined.
If $x<0$ say $x=−a$ where $a$ is a positive number,
then $[x]=−([a]+1)$.
Hence, $x^2<[x{]}^2$ and the function $f(x)=\sqrt{x^2−[x{]}^2}$ is not defined.
Therefore the domain of the function is $[0,∞)$.