What is the domain of the function \(\sec^{-1}x\) ?

Correct answer: \(( -\infty, -1] \cup [1, \infty)\)

To be defined, \(\sec^{-1}x\) must have \(\sec\theta=x\) for some angle \(\theta\). Since \(\sec\theta=\frac{1}{\cos\theta}\) and \(\cos\theta\neq 0\), we need \(\cos\theta\) to be defined and nonzero. Also for real values, \(\sec\theta\) can never have magnitude less than 1, so \(|x|\ge 1\). Therefore the domain is \(x\in(-\infty,-1]\cup[1,\infty)\).