fx=x2−x2 is defined iffx2−x2≥0⇒ x2≥x2When x is + x≥x ⇒ x2≥x2When x is − But not an integer x≤x⇒ x2≤xWhen x is − integer x=x⇒ x2=x2∴ n∈R+∪Z

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$D o m a i n o f \sqrt{x^{2} - {[x]}^{2}} i s$

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$[0, \infty)$

$R^{+} \cup Z$

$(- \infty, 0]$

$R - \{0\}$

answer is C.

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Detailed Solution

$\begin{array}{l} f (x) = \sqrt{x^{2} - {[x]}^{2}} i s d e f i n e d i f f \\ x^{2} - {[x]}^{2} \geq 0 \\ \Rightarrow x^{2} \geq {[x]}^{2} \\ W h e n x i s (+) x \geq [x] \Rightarrow x^{2} \geq {[x]}^{2} \\ W h e n x i s (-) B u t n o t a n int e g e r x \leq [x] \Rightarrow x^{2} \leq [x] \\ W h e n x i s (-) int e g e r [x] = x \Rightarrow x^{2} = {[x]}^{2} \\ ∴ n \in R^{+} \cup Z \end{array}$