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!  

Let $f(x)=\frac{x}{1+ |x|}$. Then:  $|f(x)|=\frac{|x|}{1+ |x|} \le 1$, hence
$f( \mathbb R) \subseteq [-1,1]$.
Furthermore: $\lim_{x \to \infty}f(x)=1$  and $\lim_{x \to -\infty}f(x)=-1$.
Show that $f(x) \ne 1$ and $f(x) \ne -1$ for all $x$.
Are you now in a position to derive $f( \mathbb R) =]-1,1[$ ?
