How to determine the range of the following function $\frac{x}{1+ |x|}$? How to determine the range of the following function $\frac{x}{1+ |x|}$?
when I calculated it, it was $\mathbb{R}$, but my professor said that the range is ]-1,1[, could anyone explain for me why?
thanks!  

For $x\geq 0$, $\dfrac{x}{1+|x|}=\dfrac{x}{1+x}=1-\dfrac{1}{1+x}$, it is increasing on $[0,\infty)$, so it maps onto $[0,1)$. Similarly you can deal with $(-\infty,0]$.
