Question

Find the domain of the function

f(x) = log₂(log₃(log₄(tan^–1x)²))

Answer 1

The function f is defined for 

log₃(log₄ (tan^–1x)²) > 0

⇒ (log4 (tan^–1x)²) > 1

⇒  (tan^–1x)² > 4

⇒ ((tan^–1x) + 2) ((tan^–1x)^{– 2}) > 0

⇒ x < – tan2 and x > tan2 ...(i)

Also, log₄(tan^–1x)² > 0

⇒ (tan^–1x)² > 1

⇒ ((tan^–1x) – 1) ((tan–1 x) – 1) > 0

⇒ x < –tan^-1 and x > tan^-1 ...(ii)

Again, (tan^–1x)² > 0

⇒ (tan^–1 x) > 0

⇒ x > 0 ...(iii)

From (i), (ii) and (iii), we get,

x > tan2

Thus, D_f = (tan 2, ∞)