Find the Range of \(f(x)=[x]-x\)
Question
Find the range of the function
\[ f(x)=[x]-x \]where \([x]\) denotes the greatest integer function.
Solution
Given
\[ f(x)=[x]-x \]Let
\[ x=n+t \]where \(n\in\mathbb{Z}\) and
\[ 0\le t<1 \]Then,
\[ [x]=n \]Therefore,
\[ f(x)=n-(n+t) \] \[ =-t \]Since
\[ 0\le t<1 \]multiplying by \(-1\),
\[ -1<-t\le0 \]Hence,
\[ -1