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 \(\sqrt{x}\)

For square root to be defined:

\[ x \ge 0 \] —

🔹 Step 2: Condition for \([x]\)

The greatest integer function is defined for all real numbers.

So no restriction from \([x]\).

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🔹 Step 3: Combine conditions

The only restriction is:

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

🎯 Final Answer

\[ \boxed{[0,\infty)} \]

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🚀 Exam Shortcut

• Square root ⇒ inside ≥ 0
• Greatest integer ⇒ always defined
• Domain = \([0,\infty)\)