Let f : A → R , where f x = α x 2 + 6 x - 8 α + 6 x - 8 x 2 3 . If f is onto, then the values of α lie in (given, A is the natural domain of f x )

Step-by-step solution

Let, f x = y 3 Where, y = α x 2 + 6 x - 8 α + 6 x - 8 x 2 i.e., α + 8 y x 2 + 6 1 - y x - α y + 8 = 0 According to the given condition, y takes all real values, for real x . In other words, the above quadratic equation in x should have real roots for every real y i.e., D ≥ 0 , ∀ y ∈ R i.e., 36 1 - y 2 + 4 α y + 8 α + 8 y ≥ 0 , ∀ y ∈ R i.e., 9 + 8 α y 2 + α 2 + 46 y + 9 + 8 α ≥ 0 , ∀ y ∈ R i.e., D < 0 and coefficie