NCERT Class 12 · Ch 1 Miscellaneous Exercise · Q2
NCERT Class 12 Maths · Chapter 1 Relations And Functions · Miscellaneous Exercise Question 2
Prove that f:ℝ→ℝ, f(x)=x³, is injective.
💡 Concept Hints
- Factor the difference of cubes, or use strict monotonicity of cubing on ℝ.
- Write the defining condition carefully and preserve every domain restriction while simplifying.
✅ Step-by-Step Solution
Final Answer: f is injective.
- Suppose a³=b³ for real a,b. Then (a−b)(a²+ab+b²)=0.
- The second factor equals (a+b/2)²+3b²/4≥0, and it vanishes only if a=b=0.
- If that factor vanishes then a=b; otherwise the first factor must vanish, again giving a=b. Thus equality of images forces equality of inputs.
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