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. $[0 , \infty)$
• B. $(- \infty , 0)$
• C. $(-\infty, \infty)$
• D. None of these

Answer 1

Answer: (A)

If $x \geq 0$, then $[x]$ is an integer part of $x$
Hence $x^{2} \geq[x]^{2}$ 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 $[0, \infty)$.