The function $f(x) = \dfrac{x}{1+|x|}$ maps $\mathbb{R}$ bijectively onto $(-1,1)$. It is injective (one-to-one) since $f'(x) > 0$ everywhere, and surjective onto $(-1,1)$ since $\lim_{x\to\pm\infty} f(x) = \pm 1$ (not attained). The inverse is $f^{-1}(y) = \dfrac{y}{1-|y|}$.