Class 12 Relations And Functions · Short Answer2 Markstypes_of_functions
Class 12 Maths · Relations And Functions · Short Answer #212
Prove f:R→R, f(x)=2x+sin x, is onto.
✅ Step-by-Step Solution
Final Answer: Onto R.
- For an arbitrary real y, set a=(y−1)/2 and b=(y+1)/2; f is continuous on [a,b].
- Since −1≤sin x≤1, f(a)=y−1+sin a≤y and f(b)=y+1+sin b≥y. The intermediate value theorem supplies x∈[a,b] with f(x)=y.
- Every real y has a preimage; therefore f is onto R.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] For an arbitrary real y, set a=(y−1)/2 and b=(y+1)/2; f is continuous on [a,b].
- [1 mark] Since −1≤sin x≤1, f(a)=y−1+sin a≤y and f(b)=y+1+sin b≥y. The intermediate value theorem supplies x∈[a,b] with f(x)=y.
- [0.5 marks] Every real y has a preimage; therefore f is onto R.
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