Is the Function f(x) = x/(1-x^2) Surjective on Its Domain?

[Poopsilon]

The problem is:

Prove that the function, f:(-1,1)->R given by f(x)=x/(1-x^2) is surjective

The way I have learned to prove a function is surjective is to solve for an x that is given explicitly in terms of y and then plug it into the equation and show that y=y, and thus any y in the codomain will have a corresponding x such that f(x)=y. But in this problem I can't for the life of me seem to solve for an x explicitly in terms of y, I'm assuming it can't be done. I solved for an x explicitly in terms of x and y, but I'm not sure what that gets me. There must a more intuitive, simple way to prove it.

I mean I understand why its surjective since at -1 and 1 the function diverges to -∞ and ∞ respectively so every value in R will have some corresponding input within the open interval (-1,1), but I can't figure out how to prove it without employing facts about the continuity of polynomials and limits etc. which I don't believe I am allowed to use.

 

[losiu99]

Let's try once again:
$$y=\frac{x}{1-x^2}$$
$$y - x^2y=x$$
$$y x^2 + x - y = 0$$
Now, recall the quadratic formula:
$$x=\frac{1}{2y}(1\pm \sqrt{1+4y^2})$$
for $$y\neq 0$$. But solution with "+" is clearly greater than one or less than -1.