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CBSE Class 12 Mathematics 2026 Main Paper 65/3/1 · Q325 Marksinjectivity_surjectivity

CBSE Class 12 Mathematics 2026 Main Paper 65/3/1 · Question 32 (Relations And Functions)

Show that \(f:\mathbb R\to\mathbb R\), \(f(x)=x/\sqrt{1+x^2}\), is one-one but not onto.

✅ Step-by-Step Solution

Final Answer: One-one. Its range is (−1,1), so it is not onto ℝ.
  1. The derivative f′(x)=(1+x²)^(-3/2)>0 everywhere, so f is strictly increasing and one-one.
  2. Since |x|<√(1+x²), the image lies in (−1,1). Conversely, each y in this interval has preimage x=y/√(1−y²).
  3. The range is exactly (−1,1), which is a proper subset of ℝ.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] The derivative f′(x)=(1+x²)^(-3/2)>0 everywhere, so f is strictly increasing and one-one.
  • [3 marks] Since |x|<√(1+x²), the image lies in (−1,1). Conversely, each y in this interval has preimage x=y/√(1−y²).
  • [1 mark] The range is exactly (−1,1), which is a proper subset of ℝ.

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