The range of the function y=x/(1+x^2) is

y = 0 at x = 0 and for x ≠ 0

$y=\frac{x}{1+x^2}$ or $yx^2 − x + y = 0$

$∴x=\frac{1±\sqrt{1-4y^2}}{2y}$

Since x is real, the range of the function y is determined from the relation $1-4y^2 ≥ 0, -\frac{1}{2}≤y≤\frac{1}{2}$.