Fractional Part Function

4. Fractional part function. Definition: for every real $x$, $f(x)=\{x\}$, defined as $\{x\}=x-[x]$ — whatever is left over after removing the integer (floor) part.

Illustrations: $f(4.8)=\{4.8\}=4.8-[4.8]=4.8-4=0.8$. $f(-7.1)=\{-7.1\}=-7.1-[-7.1]=-7.1-(-8)=-7.1+8=0.9$. $f(8)=\{8\}=8-[8]=8-8=0$.

Graph (Fig. 6.46): Domain $=R$; Range $=[0,1)$.

Properties: (1) If $x\in[0,1]$, $f(x)=\{x\}\in[0,1)$, shown as a slanted line segment $y=x$. At $x=0$, $f(0)=0$ and $0\in[0,1)$, shown with a filled/black disc, whereas at $x=1$, $f(1)=1$ and $1\notin[0,1)$, shown with a hollow/white disc. (2) The graph of $y=\{x\}$ lies in the region bounded by $y=0$ and $y=1$, so $0\le\{x\}<1$. (3) $\{x\}+\{-x\}=\begin{cases}0, & x\in I\\ 1, & x\notin I\end{cases}$. Example (Ex. 13): $\{5.2\}+\{-5.2\}=0.2+0.8=1$, where $5.2\notin I$; and $\{7\}+\{-7\}=0+0=0$, where $7\in I$. (4) $\{x\pm n\}=\{x\}$, where $n\in I$. Example (Ex. 14): $\{2.8+5\}=\{7.8\}=0.8$ and $\{2.8\}=0.8$; also $\{2.8-5\}=\{-2.2\}=-2.2-(-3)=0.8$ (using $\{x\}=x-[x]$ with $[-2.2]=-3$).

Ex. 15: If $\{x\}$ and $[x]$ are the fractional part function and greatest integer function of $x$ respectively, solve for $x$ if $\{x+1\}+2x=4[x+1]-6$. …