Let D be the domain of the function f x = sin - 1 log 3 x 6 + 2 log 3 x - 5 x . If the range of the function g : D → ℝ defined by g x = x - x , ( x is the greatest integer function), is α , β , then α 2 + 5 β is equal to

Step-by-step solution

We have been given f x = sin - 1 log 3 x 6 + 2 log 3 x - 5 x We know that the domain of sin - 1 x is - 1 ≤ x ≤ 1 . ⇒ log 3 x 6 + 2 log 3 x - 5 x ∈ - 1     1 - 1 ,   3 x > 0 - 2 ,   3 x ≠ 1 - 3 ,   6 + 2 log 3 x - 5 x - 4 > 0 ⇒ From 2 ,   x > 0 , from 3 ,   x ≠ 1 3 From 4 ,   6 + 2 log 3 x < 0         ∵   x > 0 ⇒ log 3 x < - 3 ⇒ x < 3 - 3 ⇒ x < 1 27 From 2