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Class 12 Maths · Relations And Functions · Long Answer #219

For A=R ∖ {2/3}, prove f:A→A, f(x)=(4x+3)/(6x−4), is bijective and find its inverse.

✅ Step-by-Step Solution

Final Answer: Bijective; f⁻¹=f.
  1. The denominator is nonzero on A; f(x)=2/3 would give 12x+9=12x−8, impossible, so f(A)⊆A.
  2. Equality f(u)=f(v) gives (4u+3)(6v−4)=(4v+3)(6u−4), hence 34(v−u)=0 and u=v.
  3. Solving y=(4x+3)/(6x−4) gives x=(4y+3)/(6y−4)=f(y); the allowed y≠2/3 gives an allowed preimage and direct substitution gives f(f(y))=y.
  4. Thus f is bijective and its inverse is itself.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] The denominator is nonzero on A; f(x)=2/3 would give 12x+9=12x−8, impossible, so f(A)⊆A.
  • [1 mark] Equality f(u)=f(v) gives (4u+3)(6v−4)=(4v+3)(6u−4), hence 34(v−u)=0 and u=v.
  • [1 mark] Solving y=(4x+3)/(6x−4) gives x=(4y+3)/(6y−4)=f(y); the allowed y≠2/3 gives an allowed preimage and direct substitution gives f(f(y))=y.
  • [0.5 marks] Thus f is bijective and its inverse is itself.

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