Plotting the function $f(x) = \frac{2x}{x^{2}+1}$ we can see the range is $[-1,1]$.

Now I was told to do $f(x)=y$ and isolate $x$ doing this I have, using the quadratic formula $x=\frac{1\pm \sqrt{1-y^2}}{y}$ then $$range f(x)= domain \frac{1\pm \sqrt{1-y^2}}{y}$$.

But this domain does not include $0$ and $0=f(0)$ so $0$ must be in the range, what am I missing?