Class 12 Relations And Functions · Short Answer2 Markstypes_of_functions
Class 12 Maths · Relations And Functions · Short Answer #167
Find the range of f(x)=4x/(3x+4), x≠−4/3, and show that f is bijective onto its range.
✅ Step-by-Step Solution
Final Answer: Range R ∖ {4/3}; f⁻¹(y)=4y/(4−3y).
- Solve y=4x/(3x+4): (4−3y)x=4y. The value y=4/3 would give the contradiction 0=16/3.
- For y≠4/3 there is a unique x=4y/(4−3y); x+4/3=16/[3(4−3y)]≠0, so it is an allowed preimage.
- Thus the range is R ∖ {4/3}, and this unique preimage gives the inverse and proves bijectivity onto that range.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Solve y=4x/(3x+4): (4−3y)x=4y. The value y=4/3 would give the contradiction 0=16/3.
- [1 mark] For y≠4/3 there is a unique x=4y/(4−3y); x+4/3=16/[3(4−3y)]≠0, so it is an allowed preimage.
- [0.5 marks] Thus the range is R ∖ {4/3}, and this unique preimage gives the inverse and proves bijectivity onto that range.
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