Question with Solution

The graph of $\mathrm{y}=\mathrm{x}-[\mathrm{x}]$ is as shown below

When x is an integer, $\mathrm{x}-[\mathrm{x}]=0$

Hence, $f(x)=0$ when $x$ is an integer

$\mathrm{x}\to [\mathrm{x}]$ as x tends to an integer.

As $\mathrm{x}\to 1,\frac{\mathrm{x}}{1+\mathrm{x}}\to \frac{1}{2}$

Hence, the range of $\mathrm{f}(\mathrm{x})$ is $\left[0,\frac{1}{2}\right)$.