Question

The range of the function $f\left(x\right)=\left[x\right]-x$ is

Answer 1

Given function is $f\left(x\right)=\left[x\right]-x$
We know that $\left[x\right]=x-\left\{x\right\}$ for $x\in R$
Where $\left\{x\right\}$is the fractional part of $x$.
$\Rightarrow f\left(x\right)=\left[x\right]-x=x-\left\{x\right\}-x$
$\Rightarrow f\left(x\right)=-\left\{x\right\}$
As we know that $0\le \left\{x\right\}<1$
$\Rightarrow -1<-\left\{x\right\}\le 0$
So,the range of $f\left(x\right)$ is$(-1,0].$
Hence, the range of the function is (-1,0]