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BITSAT2024MathematicsFunctionsActual

The function f: R R defined by f(x)= x 1+x^2 is

Options

  1. Asurjective but not injective
  2. Bbijective
  3. Cinjective but not surjective
  4. Dneither injective nor surjective

Correct answer

C. injective but not surjective

Step-by-step solution

Given that, f(x)= x 1+x^2 For injective: Let x₁, x₂ R such that f (x₁ )=f (x₂ ) x₁ 1+x₁^2 = x₂ 1+x₂^2 x₁^2 1+x₁^2 = x₂^2 1+x₂^2 x₁^2+x₁^2 x₂^2=x₂^2+x₁^2 x₂^2 x₁^2=x₂^2 x₁=x₂ So, f(x) is injective. For surjective: Let y= x 1+x^2 y^2 (1+x^2 )=x^2 y^2+y^2 x^2=x^2 x^2 (1-y^2 )=y^2 x= y^2 1-y^2 y^2 1-y^2 0 y (-1,1) So, f(x) is not surjective.

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