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f : R → R ,   f x = λ x + sin ⁡ x is onto if λ is an element of the set P and f x is one-one if λ is an element of the set Q , then (given, λ is a real number)

Options

  1. AP = Q
  2. BP ⊂ Q
  3. CP - Q = 0
  4. DQ ⊂ P

Correct answer

D. Q ⊂ P

Step-by-step solution

f ' x = λ + cos ⁡ x If f x is one-one, then λ ≥ - cos ⁡ x or λ ≤ - cos ⁡ x   ∀ x ∈ R ⇒ λ ∈ - ∞ , - 1 ∪ 1 , ∞ Further, f x has range R for λ ∈ R - 0 Hence, Q ⊂ P

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