JEE Mains · Maths · STD 12 - 1. relation and function
For the function \(f:[1,\infty) \rightarrow [1,\infty)\) defined by \(f(x)=(x-1)^4+1\), among the two statements:
(I) The set \(S=\{x \in [1,\infty): f(x)=f^{-1}(x)\}\) contains exactly two elements, and
(II) The set \(S=\{x \in [1,\infty): f(x)=f^{-1}(x+1)\}\) is an empty set,
- A only (I) is TRUE
- B only (II) is TRUE
- C both (I) and (II) are TRUE
- D neither (I) nor (II) is TRUE
Answer & Solution
Correct Answer
(A) only (I) is TRUE
Step-by-step Solution
Detailed explanation
Given \(f(x) = (x-1)^4 + 1\) for \(x \ge 1\). To find the inverse function \(f^{-1}(x)\), let \(y = (x-1)^4 + 1\). \(\Rightarrow (x-1)^4 = y-1 \Rightarrow x-1 = (y-1)^{1/4} \Rightarrow x = (y-1)^{1/4} + 1\). Thus, \(f^{-1}(x) = (x-1)^{1/4} + 1\). Evaluating the first statement,…
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