NCERT Class 12 · Ch 1 Exercise 1.2 · Q5
NCERT Class 12 Maths · Chapter 1 Relations And Functions · Exercise 1.2 Question 5
Prove that the signum function f:ℝ→ℝ, with f(x)=1 for x>0, f(0)=0 and f(x)=−1 for x<0, is neither one-one nor onto.
💡 Concept Hints
- The function is constant on each open half-line.
- Separate the one-one and onto claims. For onto, ensure each inverse candidate lies in the allowed domain.
✅ Step-by-Step Solution
Final Answer: Neither injective nor surjective.
- Two distinct positive inputs 1 and 2 have the same image 1, so f is not injective.
- The complete range of the standard signum function is {−1,0,1}.
- For example 2∈ℝ is not reached, so f is not onto ℝ.
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