Class 12 Relations And Functions · Long Answer3 Markstypes_of_functions
Class 12 Maths · Relations And Functions · Long Answer #179
For f(x)=(x−1)/(x−2), determine the domain and range and prove bijectivity onto its range.
✅ Step-by-Step Solution
Final Answer: Domain R ∖ {2}; range R ∖ {1}; f⁻¹(y)=(2y−1)/(y−1).
- The denominator requires x≠2. Writing f(x)=1+1/(x−2) shows that the output cannot be 1.
- If f(u)=f(v), equality of the nonzero reciprocals implies u=v.
- For every y≠1 choose x=(2y−1)/(y−1). Then x−2=1/(y−1)≠0 and f(x)=y, proving onto R ∖ {1}.
- Thus the stated domain and range make f bijective, with inverse (2y−1)/(y−1).
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] The denominator requires x≠2. Writing f(x)=1+1/(x−2) shows that the output cannot be 1.
- [1 mark] If f(u)=f(v), equality of the nonzero reciprocals implies u=v.
- [1 mark] For every y≠1 choose x=(2y−1)/(y−1). Then x−2=1/(y−1)≠0 and f(x)=y, proving onto R ∖ {1}.
- [0.5 marks] Thus the stated domain and range make f bijective, with inverse (2y−1)/(y−1).
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