Question

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

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

Answer 1

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

We know that $\left[x\right]$ is the greatest integer function which gives the integral part of the real number $x$.

Now we know that any real number consists of two parts, integral part which is $\left[x\right]$ and the fractional part given by $\left\{x\right\}$.

$⟹x=\left[x\right]+\left\{x\right\}$

$⟹x-\left[x\right]=\left\{x\right\}$

And, the fractional part of any real number is $\left\{x\right\}\in [0,1)$

$⟹f\left(x\right)=x-\left[x\right]\in [0,1)$

Hence , range of function is $[0,1)$.