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MHT CET202520 Apr 2025Morning ShiftMathematicsFunctionsActual

The function defined by f (x)= 2 x+3 3 x+4 , x - 4 3 is

Options

  1. Aonly one one
  2. Bonly onto
  3. Conto for y 2 3 and one - one
  4. Dneither one-one nor onto

Correct answer

C. onto for y 2 3 and one - one

Step-by-step solution

To test the injectivity of f(x) = 2x + 3 3x + 4 , where x - 4 3 , suppose f(x₁) = f(x₂) . This implies 2x₁ + 3 3x₁ + 4 = 2x₂ + 3 3x₂ + 4 . Cross-multiplying gives (2x₁ + 3)(3x₂ + 4) = (2x₂ + 3)(3x₁ + 4) . Expanding both sides yields 6x₁x₂ + 8x₁ + 9x₂ + 12 = 6x₁x₂ + 8x₂ + 9x₁ + 12 . Subtracting 6x₁x₂ + 12 from both sides gives 8x₁ + 9x₂ = 8x₂ + 9x₁ , which simplifies to x₁ = x₂ , confirming the function is one-one. To analyze surjectivity, let y = f(x) and solve for x : y(3x + 4) = 2x + 3 , so 3xy + 4y = 2x + 3 . Re

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