Question

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

• A. $R$
• B. $[0,+\infty )$
• C. $(-\infty ,0]$
• D. $x\in [0,+\infty ]\cup$ [Negative integers ]

Answer 1

The correct option is D $x\in [0,+\infty ]\cup$ [Negative integers ]

Domain of function $f\left(x\right)=\sqrt{x^2-[x{]}^2}$

(1) when $x$ is positive

$x\ge \left[x\right]$

then $x^2-[x{]}^2\ge 0$

(2) when $x$ is negative but not an integer

$x\le \left[x\right]$

then $x^2-[x{]}^2\le 0$

(3) when $x$ is negative and integer points

$x^2-[x{]}^2$ is zero

So, Domain $x\in [0,+\infty ]\cup$ [Negative integers ]

$D$ is correct