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JEE Advanced2020MathematicsFunctionsActual

If the function f : R → R is defined by f x = x x − sin x , then which of the following statements is TRUE?

Options

  1. Af is one-one, but NOT onto
  2. Bf is onto, but NOT one-one
  3. Cf is BOTH one-one and onto
  4. Df is NEITHER one-one NOR onto

Correct answer

C. f is BOTH one-one and onto

Step-by-step solution

f x = x x − sin x is odd function ∵ f − x = − f x Now, f x = x 2 − x sin x ∀ x ≥ 0 ⇒ f ' x = 2 x − x cos x − sin x ⇒ f ' x = x − sin x + x 1 − cos x > 0 ∴ Graph of y = f x will be From graph, it's one-one and onto both.

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