For \(x\ne0\), the graph of the function \(f(x)=\frac{|x|}{x}\) is the same as the graph of which standard function?

Correct answer: \(f(x)=\operatorname{sgn}(x)\)

For \(x>0\), \(|x|=x\), so \(\frac{|x|}{x}=1\). For \(x<0\), \(|x|=-x\), so \(\frac{|x|}{x}=-1\). Therefore, its graph is the same as \(y=\operatorname{sgn}(x)\) on the domain \(x\ne0\); the function is not defined at \(x=0\). Option B reverses both values, while options C and D vary continuously with \(x\). Exam tip: split expressions containing \(|x|\) into the cases \(x>0\) and \(x<0\).