Find Domain of \(f(x)=\frac{1}{\sqrt{[x]-x}}\)

📝 Question

Find the domain of the function:

\[ f(x)=\frac{1}{\sqrt{[x]-x}} \]

where \([x]\) denotes the greatest integer function.

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✅ Solution

🔹 Step 1: Conditions for denominator

For the function to be defined:

• \([x]-x \ge 0\) (inside square root)
• \([x]-x \ne 0\) (denominator cannot be zero)

So,

\[ [x]-x > 0 \] —

🔹 Step 2: Use property of greatest integer function

We know:

\[ [x] \le x < [x]+1 \]

Thus,

\[ [x]-x \le 0 \] —

🔹 Step 3: Compare conditions

We need:

\[ [x]-x > 0 \]

But from property:

\[ [x]-x \le 0 \]

This is not possible.

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🎯 Final Answer

\[ \boxed{\varnothing} \]

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🚀 Exam Shortcut

• \([x]-x \le 0\) always
• But need \(>0\) for denominator
• Impossible ⇒ no real solution