Let f : R - - 1 2 → R and g : R - - 5 2 → R be defined as f x = 2 x + 3 2 x + 1 and g x = | x | + 1 2 x + 5 . Then the domain of the function fog is :

Step-by-step solution

Given: g x = x + 1 2 x + 5 . Also, g x ≠ - 1 2 ⇒ x + 1 2 x + 5 ≠ - 1 2 Case-I: x ≥ 0 ⇒ g x = x + 1 2 x + 5 = - 1 2 ⇒ 2 x + 2 = - 2 x - 5 ⇒ 4 x = - 7 ⇒ x = - 7 4 → Rejected as x ≥ 0 Case-I: x < 0 ⇒ - x + 1 2 x + 5 = - 1 2 ⇒ - 2 x + 2 = - 2 x - 5 ⇒ 2 = - 5 → Not possible So, domain of f g x = domain of g x . ⇒ Domain of fog x = R - - 5 2