For the function , with the rule , find the rule for the inverse function.
note that f(t) will need to be restricted to the domain $\displaystyle (-\infty,-1]$ in order to have an inverse function.
$\displaystyle y = \frac{3t}{t^2+1}$
$\displaystyle t = \frac{3y}{y^2+1}$
$\displaystyle t(y^2+1) = 3y$
$\displaystyle ty^2 + t = 3y$
$\displaystyle ty^2 - 3y + t = 0$
$\displaystyle y = \frac{3 \pm \sqrt{9 - 4t^2}}{2t}$
$\displaystyle f^{-1}(t) = \frac{3 +\sqrt{9 - 4t^2}}{2t} \, ; \, t \in [-1.5,0)$