Question

The range of the function $f\left(x\right)=x-\left[x\right]$ where $\left[x\right]$ represents the greatest integer less than or equal to $x$ is

• A. $\left\{0\right\}$
• B. $[0,1]$
• C. $(0,1)$
• D. $[0,1)$

Answer 1

The correct option is D $[0,1)$

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

$f\left(x\right)=x-\left[x\right]=\left\{x\right\}\Rightarrow$ fractional part of a integer

It is well known that the fractional part function always between $0$ and $1$ and $\left\{x\right\}$ can also be zero

$\Rightarrow 0\le \left\{x\right\}<1$

$\therefore$ the range of f(x) is $[0,1)$