Find the Domain and Range of the Function
Question:
Write the domain and range of the function
\[ f(x)=\sqrt{[x]-x} \]
Solution:
Since
\[ [x]\le x<[x]+1 \]
therefore
\[
-1
Multiplying by \(-1\),
\[
0\le [x]-x<1
\]
Hence square root is defined for all real \(x\).
Domain:
\[
\boxed{R}
\]
Also,
\[
0\le [x]-x<1
\]
Taking square root,
\[
0\le \sqrt{[x]-x}<1
\]
Therefore, range is
\[
\boxed{[0,1)}
\]