I've formulated a proof, but also a counterexample?

Question

show that f(x) is injective.

$f(x)= \frac{x^2}{1+x^2}$

if $f(x)=f(y)$ then $\frac{x^2}{1+x^2}=\frac{y^2}{1+y^2}$

$(x^2)(1+y^2)=(y^2)(1+x^2)$

$x^2+x^2y^2=y^2+x^2y^2$

$x^2 = y^2$

$x = y$

but $f(1)=f(-1)$ due to the square roots. Where did I go wrong in the original proof?

Answer 1

If $x^2=y^2$ then $x=y$ or $x=-y$.
You suppose only $x=y$.
There is the mistake.

Answer 2

$x^2 = y^2$ does not imply $x = y$.

$x^2 = y^2 \Leftrightarrow (x = y \mbox{ or } x = -y)$.