Question

The domain of the function f(x)=√x²-[x]², where [x] denotes the greatest integer less than or equal to x, is:

• A. \( (0,\infty) \)
• B. \( (-\infty,0) \)
• C. \( (-\infty,\infty) \)
• D. None of these

Hint:

Use x=[x]+x with 0≤x<1 for greatest integer function problems.

Answer 1

The Correct Option is C

Step 1: For any real x, write x=[x]+x where 0≤ x<1.
Step 2: Then x²-[x]² = ([x]+x)²-[x]² = 2[x]x+x² ≥ 0.
Step 3: Hence the expression under the square root is non-negative for all real x.