If \(f(x)=\sqrt{x-1}\) and \(g(x)=x-5\), what is the domain of \(\left(\frac{f}{g}\right)(x)\)?

Correct answer: \([1,\infty)\setminus\{5\}\)

For \(\sqrt{x-1}\) to be defined, \(x-1\ge 0\), so \(x\ge 1\). In addition, the denominator \(g(x)=x-5\) must be nonzero, so \(x\ne 5\). Combining these conditions gives the domain \([1,\infty)\setminus\{5\}\). Option B is incorrect because \(x=1\) is allowed: the square root is \(0\) there. Exam tip: for a quotient, intersect the domains of the numerator and denominator functions, then exclude every value that makes the denominator zero.