<s>[INST] How can I prove that $\frac{x}{x^2 - 4}$is invertible? 
Prove that $$f(x)=\frac{x}{x^2-4}$$ is invertible in $(-2,2)$.

I´m trying to proof this using the definition of injective function:
$$\frac{x}{x^2-4}=\frac{y}{y^2-4} \rightarrow x=y$$ but I couldn´t prove this definition.
 [/INST] For $x$ and $y$ in our interval, we have
$$\frac{x}{x^2-4}=\frac{y}{y^2-4}\quad\text{if and only if}\quad x(y^2-4)=y(x^2-4).$$
This can be rewritten as $xy^2-yx^2+4(y-x)=0$, and then as
$$(xy+4)(y-x)=0.$$
In our interval, we cannot have $xy+4=0$, so $y=x$.
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