JEE Mains · Maths · STD 12 - 1. relation and function
Consider function \(f: A \rightarrow B\) and \(g: B \rightarrow C(A, B, C \subseteq R)\) such that \((gof) ^{-1}\) exists, then:
- A \(\mathrm{f}\) and \(\mathrm{g}\) both are one-one
- B \(\mathrm{f}\) is onto and \(\mathrm{g}\) is one-one
- C \(\mathrm{f}\) is one-one and \(\mathrm{g}\) is onto
- D \(\mathrm{f}\) and \(\mathrm{g}\) both are onto
Answer & Solution
Correct Answer
(C) \(\mathrm{f}\) is one-one and \(\mathrm{g}\) is onto
Step-by-step Solution
Detailed explanation
\(\therefore(\text { gof })^{-1}\) exist \(\Rightarrow\) gof is bijective \(\Rightarrow g f(x)\) must be bijective. \(\Rightarrow^{\prime} \mathrm{f}^{\prime}\) must be one-one and \('^{\prime}\mathrm{g}^{\prime}\) must be \(ONTO\)
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