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Thread: Inverse function

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    Inverse function

    For the function , with the rule , find the rule for the inverse function.
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    Quote Originally Posted by usagi_killer View Post
    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.

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

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

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

    $\displaystyle ty^2 + t = 3y$

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

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


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