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 ∩ B = (1, ∞) – N and  
(S2) : A ∪ B = (1, ∞) 

• A. (1) only (S1) is true
• B. both (S1) and (S2) are true
• C. neither (S1) nor (S2) is true
• D. only (S2) is true

Answer 1

The Correct Option is A

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.