CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q222 Markssurjectivity
CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Question 22 (Relations And Functions)
Is f: ℝ∖{3}→ℝ, f(x)=(x−2)/(x−3), onto? OR Is f: ℤ×ℤ→ℤ×ℤ, f(x,y)=(2y,3x), injective?
Alternative 1 2 Marks
Is f: ℝ∖{3}→ℝ, f(x)=(x−2)/(x−3), onto?
Answer: No. Its range is ℝ∖{1}.
- Set y=(x−2)/(x−3).
- At y=1 the equation would give −2=−3. For y≠1 it gives x=(3y−2)/(y−1), which is never 3.
- Thus 1 has no preimage and f is not onto.
Alternative 2 2 Marks
Is f: ℤ×ℤ→ℤ×ℤ, f(x,y)=(2y,3x), injective?
Answer: Yes.
- Suppose f(x₁,y₁)=f(x₂,y₂).
- Equality of coordinates gives 2y₁=2y₂ and 3x₁=3x₂.
- Therefore both input coordinates agree and f is injective.