For which real values of \(x\) is \(\cosec^{-1}(x-2)\) defined?

Correct answer: \(x\le 1\) or \(x\ge 3\)

For \(\cosec^{-1}(u)\) to be defined, we need \(|u|\ge 1\) because \(\csc\theta = u\) requires \(u\neq 0\) and \(u\) must lie in the domain of inverse cosecant. Here \(u=x-2\), so we need \(|x-2|\ge 1\). This gives \(x-2\ge 1\Rightarrow x\ge 3\) or \(x-2\le -1\Rightarrow x\le 1\). Hence \(x\le 1\) or \(x\ge 3\).