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JEE Main20266 April 2026Evening ShiftMathematicsFunctionsActual

Let f: R R be defined as f(x) = 2x^2 - 3x + 2 3x^2 + x + 3 . Then f is :

Options

  1. Aboth one-one and onto
  2. Bone-one but not onto
  3. Conto but not one-one
  4. Dneither one-one nor onto

Correct answer

D. neither one-one nor onto

Step-by-step solution

Given f(x) = 2x^2 - 3x + 2 3x^2 + x + 3 To check for one-one, let f(x) = f(y) 2x^2 - 3x + 2 3x^2 + x + 3 = 2y^2 - 3y + 2 3y^2 + y + 3 Cross-multiplying and simplifying, we get: 11x^2y - 11xy^2 - 11x + 11y = 0 11xy(x - y) - 11(x - y) = 0 11(x - y)(xy - 1) = 0 This implies x = y or xy = 1 . Since f(x) = f(1/x) for x 0 (for example, f(2) = f(1/2) = 4 17 ), the function is many-one (not one-one). To check for onto, let y = 2x^2 - 3x + 2 3x^2 + x + 3 (3y - 2)x^2 + (y + 3)x + (3y - 2) = 0 For x R , the discriminant D 0 :

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