If \(f(x)=\sqrt{x-1}\) and \(g(x)=x-3\), what is the domain of the quotient function \(\left(\frac{g}{f}\right)(x)\)?

Correct answer: \((1,\infty)\)

For \(f(x)=\sqrt{x-1}\), we need \(x\ge 1\). However, in the quotient \(\frac{x-3}{\sqrt{x-1}}\), the denominator must not be zero. At \(x=1\), the denominator is zero, so the actual condition is \(x>1\), giving the domain \((1,\infty)\). Option B incorrectly includes \(x=1\). Exam tip: For a quotient involving a square root, apply both the square-root condition and the non-zero-denominator condition.