Showing function is injective? I want to show that this function is injective -
$f(x) = \frac{x}{1 - x^2}$
So when $f(x) = f(y)$ I should have $x = y$
$\frac{y}{1 - y^2} = \frac{x}{1 - x^2}$
$x - xy^2 = y - x^2y$
$x - y = xy^2 - x^2y$
$x - y = xy(y - x)$
$\frac{x-y}{y-x} = xy$
$\frac{x-y}{x-y} = -xy$
$-xy = 1$
$x = -\frac{1}{y}$
Where am I going wrong?

$x - y = xy(y - x)$
$(x-y)(1+xy)=0$
$x=y$ or $x=-1/y$.
So the function is not injective.
