Class 12 Relations And Functions · Long Answer3 Markstypes_of_functions
Class 12 Maths · Relations And Functions · Long Answer #220
For N={1,2,…}, prove f:N→N, f(n)=n²+n+1, is one-one but not onto.
✅ Step-by-Step Solution
Final Answer: One-one but not onto.
- The formula gives positive integers for all positive integer inputs.
- If f(n)=f(m), then (n−m)(n+m+1)=0. Since n+m+1>0, n=m, proving injectivity.
- For every n≥1, f(n)≥3; hence the codomain element 2 has no preimage, disproving surjectivity.
- Therefore f is one-one but not onto N.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] The formula gives positive integers for all positive integer inputs.
- [1 mark] If f(n)=f(m), then (n−m)(n+m+1)=0. Since n+m+1>0, n=m, proving injectivity.
- [1 mark] For every n≥1, f(n)≥3; hence the codomain element 2 has no preimage, disproving surjectivity.
- [0.5 marks] Therefore f is one-one but not onto N.
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