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Let the function f: [2, ) [ 1 3 , 1 ) be defined as f(x) = x^2 - 2x + 4 x^2 + 2x + 4 . Then f is :

Options

  1. Aneither one-one nor onto
  2. Bone-one but not onto
  3. Cboth one-one and onto
  4. Donto but not one-one

Correct answer

C. both one-one and onto

Step-by-step solution

Given f(x) = x^2 - 2x + 4 x^2 + 2x + 4 . To check for one-one, we find the derivative of f(x) with respect to x : f'(x) = (2x - 2)(x^2 + 2x + 4) - (x^2 - 2x + 4)(2x + 2) (x^2 + 2x + 4)^2 f'(x) = 2(x - 1)(x^2 + 2x + 4) - 2(x + 1)(x^2 - 2x + 4) (x^2 + 2x + 4)^2 f'(x) = 2(x^3 + x^2 + 2x - 4) - 2(x^3 - x^2 + 2x + 4) (x^2 + 2x + 4)^2 f'(x) = 2(2x^2 - 8) (x^2 + 2x + 4)^2 = 4(x^2 - 4) (x^2 + 2x + 4)^2 For x > 2 , x^2 - 4 > 0 , which implies f'(x) > 0 . Thus, f(x) is strictly increasing on [2, ) , meaning f is a one-one (i

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