Question

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

• A. $f\left(x\right)=1,2,3\dots$
• B. $f\left(x\right)\ge 0$
• C. $f\left(x\right)<1$
• D. $0\le f\left(x\right)<1$

Answer 1

The correct option is D $0\le f\left(x\right)<1$

$f\left(x\right)=x-\left[x\right]$

$\Rightarrow f\left(x\right)=\{\begin{array}{c}0x\in Z\\ \left\{x\right\}x\notin Z\end{array}$

where $\left\{x\right\}$ is the fractional part of $x$.

The range of $\left\{x\right\}$ is $[0,1)$.

Hence, the range of $f\left(x\right)$ is $[0,1)$.

$\therefore 0\le f\left(x\right)<1$