# Math Help - Inverse function

1. ## Inverse function

For the function $f:(-\infty,0] \to R$, with the rule $f(t) = \frac{3t}{t^2+1}$, find the rule for the inverse function.

2. Originally Posted by usagi_killer
For the function $f:(-\infty,0] \to R$, with the rule $f(t) = \frac{3t}{t^2+1}$, find the rule for the inverse function.
note that f(t) will need to be restricted to the domain $(-\infty,-1]$ in order to have an inverse function.

$y = \frac{3t}{t^2+1}$

$t = \frac{3y}{y^2+1}$

$t(y^2+1) = 3y$

$ty^2 + t = 3y$

$ty^2 - 3y + t = 0$

$y = \frac{3 \pm \sqrt{9 - 4t^2}}{2t}$

$f^{-1}(t) = \frac{3 +\sqrt{9 - 4t^2}}{2t} \, ; \, t \in [-1.5,0)$