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Class 12 Relations And Functions · Long Answer3 Markstypes_of_functions

Class 12 Maths · Relations And Functions · Long Answer #178

Prove f:R ∖ {3}→R ∖ {1}, f(x)=(x−2)/(x−3), is bijective.

✅ Step-by-Step Solution

Final Answer: Bijective; f⁻¹(y)=(3y−2)/(y−1).
  1. For x≠3, f(x)=1+1/(x−3), so f(x)≠1 and the map is well defined.
  2. Equality f(u)=f(v) implies 1/(u−3)=1/(v−3), hence u=v, proving injectivity.
  3. For any y≠1 take x=(3y−2)/(y−1); then x−3=1/(y−1)≠0 and substitution gives f(x)=y.
  4. Therefore f is onto as well as one-one, and its inverse is (3y−2)/(y−1).

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] For x≠3, f(x)=1+1/(x−3), so f(x)≠1 and the map is well defined.
  • [1 mark] Equality f(u)=f(v) implies 1/(u−3)=1/(v−3), hence u=v, proving injectivity.
  • [1 mark] For any y≠1 take x=(3y−2)/(y−1); then x−3=1/(y−1)≠0 and substitution gives f(x)=y.
  • [0.5 marks] Therefore f is onto as well as one-one, and its inverse is (3y−2)/(y−1).

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