Question:

Let the sets A and B denote the domain and range respectively of the function f(x)=1βˆ’xxf(x)=\frac{1}{\sqrt[x]-x} where [x] denotes the smallest integer greater than or equal to x . Then among the statements 
(S1) : A ∩ B = (1, ∞) – N and  
(S2) : A βˆͺ B = (1, ∞) 

Updated On: Mar 21, 2025
  • (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 x > 1 but excludes natural numbers N \mathbb{N} from the domain due to the square root denominator constraint.

Since A∩B=(1,∞)βˆ’N A \cap B = (1, \infty) - \mathbb{N} , (S1) is true.

For (S2), AβˆͺB A \cup B does not cover all real values in (1,∞) (1, \infty) , making (S2) false.

Final Answer: (A) Only (S1) is true.

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