If f tan ⁡ x = sin ⁡ 2 x : x ≠ 2 n + 1 π 2 , n ∈ Ι , then which of the following is an incorrect statement?

Question

If f tan ⁡ x = sin ⁡ 2 x : x ≠ 2 n + 1 π 2 , n ∈ Ι , then which of the following is an incorrect statement?, Domain of f x is R - 2 n + 1 π 2 , n ∈ Ι, Range of f x is - 1,1, f x is odd function, f x is many-one function

MARKS App · Accepted Answer

f tan ⁡ x = 2 tan ⁡ x 1 + tan 2 ⁡ x ⇒ f x = 2 x 1 + x 2 Now, domain of f x is x ∈ R For the range of f x ⇒ y = 2 x 1 + x 2 ⇒ y x 2 - 2 x + y = 0 ∵ x ∈ R ∴ D ≥ 0 4 - 4 y 2 ≥ 0 ⇒ y ∈ - 1,1 is the range Also, f - x = 2 - x 1 + - x 2 = - f x ∴ f x is an odd function f ' x = 1 + x 2 2 - 2 x 2 x 1 + x 2 2 = 2 1 - x 2 1 + x 2 2 f ' x can be positive & negative ∀ x ∈ R ∴ f x is many-one

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