Question

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

• A. $(-1,0]$
• B. $(-1,1]$
• C. $(1,0)$
• D. $(1,2]$

Answer 1

The correct option is A $(-1,0]$

We have,
$\left[x\right]=x-\left\{x\right\}$ for $x\in R.$ where $\left\{x\right\}$ is the fractional part of $x.$
Now,
$\left[x\right]-x=-\left\{x\right\}.$
As we have $0\le \left\{x\right\}<1$
$\Rightarrow -1<-\left\{x\right\}\le 0.$
So, range of $f\left(x\right)$ is $(-1,0].$