Let $f\left(x\right)$ $=\frac{x-\left[x\right]}{1-\left[x\right]+x}$ , then range of $f\left(x\right)$  is ($\left([.]=G.I.F.\right)$

$y=f\left(x\right)$ $=\frac{x-\left[x\right]}{1-\left[x\right]+x},[.]=G.I.F.$

$\Rightarrow y=\frac{\left\{x\right\}}{1+\left\{x\right\}}\Rightarrow y$ $+y\left\{x\right\}=\left\{x\right\}$

$\Rightarrow \left\{x\right\}$ $=\frac{y}{1-y}\left(\{.\}=fractionalpart\right)$

$\Rightarrow 0\le \frac{y}{1-y}<1\Rightarrow 0\le y\le 1/2$

Hence, range $f\left(x\right)=\left[0,1/2\right)$ .