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CBSE Class 12 Mathematics 2026 Main Paper 65/3/2 · Q345 Marksinjectivity_surjectivity

CBSE Class 12 Mathematics 2026 Main Paper 65/3/2 · Question 34 (Relations And Functions)

Show that f:[0,∞)→[−5,∞), f(x)=4x²+4x−5 is one-one and onto. Find p≥0 with f(p)=3.

✅ Step-by-Step Solution

Final Answer: One-one and onto. p=1.
  1. On x≥0, f′=8x+4>0, proving injectivity.
  2. Since f(x)=(2x+1)²−6, each y≥−5 has a nonnegative preimage (√(y+6)−1)/2, proving surjectivity.
  3. For y=3 this preimage is p=1.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] On x≥0, f′=8x+4>0, proving injectivity.
  • [3 marks] Since f(x)=(2x+1)²−6, each y≥−5 has a nonnegative preimage (√(y+6)−1)/2, proving surjectivity.
  • [1 mark] For y=3 this preimage is p=1.

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