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Let a function f : 2 , ∞ → 0 , ∞ defined as f x = x - 3 x - 2 , then f is

Options

  1. Ainjective & surjective
  2. Bnot injective but surjective
  3. Cinjective but not surjective
  4. Dneither injective nor surjective

Correct answer

B. not injective but surjective

Step-by-step solution

f ( x ) = 1 - 1 x - 2 ∵ 2 < x < ∞ 0 < x - 2 < ∞ ⇒ ∞ > 1 x - 2 > 0 - ∞ < - 1 x - 2 < 0 ⇒ - ∞ < 1 - 1 x - 2 < 1 ∵ 0 ≤ 1 - 1 x - 2 < ∞ ∴ Range of f x ∈ 0 , ∞ = co-domain Hence, f ( x ) is surjective Let, f x = 1 2 ⇒ 1 - 1 x - 2 = 1 2 1 - 1 x - 2 = 1 2 ,   1 - 1 x - 2 = - 1 2 1 x - 2 = 1 2 ,   1 x - 2 = 3 2 x = 4 ,   x = 8 3 ∴ f 4 = f 8 3 = 1 2 ∴ f x is many-on

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