# Find the domain of

Question:

Find the domain of

(i) $\sec ^{-1}(3 x-1)$

(ii) $\sec ^{-1} x-\tan ^{-1} x$

Solution:

(ii)

Let $f(x)=g(x)-h(x)$, where

Therefore, the domain of $f(x)$ is given by the intersection of the domain of $g(x)$ and $h(x)$

The domain of $g(x)$ is $[0, \pi / 2) \cup[\pi, 3 \pi / 2)$

The domain of $h(x)$ is $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

Therfore, the intersection of $g(x)$ and $h(x)$ is $\mathrm{R}-\{n \pi, n \in Z\}$