Class 12 Relations And Functions · Short Answer2 Markstypes_of_functions
Class 12 Maths · Relations And Functions · Short Answer #166
Prove f:[0,∞)→[4,∞), f(x)=x²+4, is invertible and find its inverse.
✅ Step-by-Step Solution
Final Answer: f⁻¹(y)=√(y−4), y≥4.
- If f(u)=f(v), then (u−v)(u+v)=0; nonnegative u,v imply u=v.
- For any y≥4 choose x=√(y−4)≥0; f(x)=y, proving onto as well as one-one.
- Hence f is invertible, with f⁻¹(y)=√(y−4) on [4,∞).
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] If f(u)=f(v), then (u−v)(u+v)=0; nonnegative u,v imply u=v.
- [1 mark] For any y≥4 choose x=√(y−4)≥0; f(x)=y, proving onto as well as one-one.
- [0.5 marks] Hence f is invertible, with f⁻¹(y)=√(y−4) on [4,∞).
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