How do you work out the inverse of functions such as $ f(x)= \frac{x}{ x^2-1} $?

Question

how do you find the inverse of a function such as:

$$f(x) = \frac{x}{x^2-1} , x \in (-1,1)$$

Answer 1

In general, to find the inverse of $f(x)$, one solves $x = f(y)$ for $y$. In your case that yields $$ x = \frac{y}{y^2-1} \Leftrightarrow xy^2 -y - x = 0 $$ for which you can apply the quadratic formula $$ y_\pm = \frac{-1 \pm \sqrt{1-4x^2}}{2x} $$ So there are 2 inverse functions.

Answer 2

All you have to do is to solve the equation $\frac x{x^2-1}=a$ and to find a solution in $(-1,1)$.