NCERT Class 12 · Ch 1 Exercise 1.2 · Q7
NCERT Class 12 Maths · Chapter 1 Relations And Functions · Exercise 1.2 Question 7
Classify (i) f:ℝ→ℝ, f(x)=3−4x; (ii) f:ℝ→ℝ, f(x)=1+x², as one-one, onto or bijective, with justification.
💡 Concept Hints
- An affine function with nonzero slope can be inverted; an even power may identify opposite inputs.
- Separate the one-one and onto claims. For onto, ensure each inverse candidate lies in the allowed domain.
✅ Step-by-Step Solution
Final Answer: (i) bijective; (ii) neither one-one nor onto.
- (i) If 3−4a=3−4b, then a=b; therefore injective.
- For every y∈ℝ, x=(3−y)/4 is real and gives f(x)=y; therefore onto and hence bijective.
- (ii) f(1)=f(−1)=2, so not injective. Also f(x)≥1 for all x, and every value ≥1 is reached.
- Thus the second range is [1,∞), not all of ℝ; it is not onto or bijective.
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