What is the domain of the real-valued function \(f(x)=\sqrt{|x|-3}\)?

Correct answer: \(( -\infty,-3]\cup[3,\infty)\)

For a real square root, the expression under the radical must be non-negative. Hence \(|x|-3\ge 0\), so \(|x|\ge 3\). Therefore, \(x\le -3\) or \(x\ge 3\), giving the domain \(( -\infty,-3]\cup[3,\infty)\). Option B wrongly includes values between \(-3\) and \(3\), where \(|x|-3<0\). Exam tip: For real functions involving a square root, first set the radicand greater than or equal to zero.