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, Df = (tan 2, ∞)