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

📝 Question

Find the domain of the function:

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

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

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✅ Solution

🔹 Step 1: Condition for square root

For the function to be defined:

\[ [x]-x \ge 0 \] —

🔹 Step 2: Use property of greatest integer function

We know:

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

So,

\[ [x]-x \le 0 \] —

🔹 Step 3: Combine conditions

We need:

\[ [x]-x \ge 0 \]

But we already have:

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

Thus,

:contentReference[oaicite:0]{index=0} —

🔹 Step 4: Solve

\[ [x]=x \]

This happens only when \(x\) is an integer.

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🎯 Final Answer

\[ \boxed{\mathbb{Z}} \]

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🚀 Exam Shortcut

• \([x] \le x\) always
• So \([x]-x \le 0\)
• For square root ⇒ must be 0
• Thus \(x\) must be integer