Question:
Let the sets A and B denote the domain and range respectively of the function \(f(x)=\frac{1}{\sqrt[x]-x}\) where [x] denotes the smallest integer greater than or equal to x . Then among the statements
(S1) : A ā© B = (1, ā) ā N and
(S2) : A āŖ B = (1, ā)
Let the sets A and B denote the domain and range respectively of the function \(f(x)=\frac{1}{\sqrt[x]-x}\) where [x] denotes the smallest integer greater than or equal to x . Then among the statements
(S1) : A ā© B = (1, ā) ā N and
(S2) : A āŖ B = (1, ā)
(S1) : A ā© B = (1, ā) ā N and
(S2) : A āŖ B = (1, ā)
Updated On: Mar 26, 2026
- (1) only (S1) is true
- both (S1) and (S2) are true
- neither (S1) nor (S2) is true
- only (S2) is true
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The Correct Option is A
Solution and Explanation
Solution:
The function is defined for \( x > 1 \) but excludes natural numbers \( \mathbb{N} \) from the domain due to the square root denominator constraint.
Since \( A \cap B = (1, \infty) - \mathbb{N} \), (S1) is true.
For (S2), \( A \cup B \) does not cover all real values in \( (1, \infty) \), making (S2) false.
Final Answer: (A) Only (S1) is true.
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