Question

Find the domain of
(i) ${\sec }^{-1}\left(3x-1\right)$
(ii) ${\sec }^{-1}x-{\tan }^{-1}x$

Answer 1

(ii)

Let f(x) = g(x) − h(x), where g(x)=cotx and h(x)=cot−1x
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, π/2) ⋃ [ π, 3π/2)
The domain of h(x) is $\left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$
Therfore, the intersection of g(x) and h(x) is R − { nπ, n ⋵ Z}