Question

Find the range of $f\left(x\right)=\left[x\right]-x$. Where $\left[x\right]$ is the greatest integer function.

Answer 1

We've,

$\left[x\right]=x-\left\{x\right\}$ for $x\in R$. Where $\left\{x\right\}$ is the fractional part of $x$.

Now,

$\left[x\right]-x=-\left\{x\right\}$.

As we've $0\le \left\{x\right\}<1$

or, $-1<-\left\{x\right\}\le 0$.

So range of of $f\left(x\right)$ is $(-1,0]$.