NCERT Class 12 · Ch 1 Miscellaneous Exercise · Q1
NCERT Class 12 Maths · Chapter 1 Relations And Functions · Miscellaneous Exercise Question 1
Prove that f:ℝ→(−1,1), f(x)=x/(1+|x|), is one-one and onto.
💡 Concept Hints
- Split by the sign of x; on each branch solve the output equation.
- Write the defining condition carefully and preserve every domain restriction while simplifying.
✅ Step-by-Step Solution
Final Answer: f is bijective; f⁻¹(y)=y/(1−|y|), −1<y<1.
- For every x, |f(x)|=|x|/(1+|x|)<1, so the stated codomain is valid. The sign of a nonzero output equals the input sign.
- For y∈[0,1), solve y=x/(1+x) with x≥0 to obtain x=y/(1−y). This is the unique nonnegative solution.
- For y∈(−1,0), solve y=x/(1−x) with x<0 to obtain x=y/(1+y). This is the unique negative solution.
- The two formulas combine to x=y/(1−|y|). Every y∈(−1,1) has exactly one preimage, proving both onto and one-one.
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