Let f : - 1 , 1 → B be a function defined as f x = cot - 1 ⁡ 2 x 3 1 + x 2 . If f is both one-one and onto, then B is the interval

Question

Let f : - 1 , 1 → B be a function defined as f x = cot - 1 ⁡ 2 x 3 1 + x 2 . If f is both one-one and onto, then B is the interval, 0 , x 3, 0 , 2 π 3, π 3 , 2 π 3, π 3 , π

MARKS App · Accepted Answer

Let, y = 2 x 3 1 + x 2 ⇒ 3 y x 2 - 2 x + 3 y = 0 ∵ x ∈ R ⇒ D ≥ 0 ⇒ 4 - 12 y 2 ≥ 0 ⇒ y ∈ - 1 3 , 1 3 ⇒ - 1 3 ≤ 2 x 3 1 + x 2 ≤ 1 3 ⇒ cot - 1 ⁡ - 1 3 ≥ cot - 1 ⁡ 2 x 3 1 + x 2 ≥ cot - 1 ⁡ 1 3 2 π 3 ≥ f x ≥ π 3

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