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Math Help - Inverse function Help

  1. #1
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    Inverse function Help

    For this problem, y = f(x) = 3x2 − 5x − 1
    on the domain of all real numbers.

    1. Restrict
    y = f(x)to the domain {ax} and find the formula for the inverse function f−1(y).

    I found the vertex (5/6, -37/12)
    But I am not sure what to do now for the problem. Any help is appreciated, thank you!
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  2. #2
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    Re: Inverse function Help

    x = 3y^2 - 5y - 1 , x \ge \frac{5}{6}

    x = 3 \left(y^2 - \frac{5}{3}y \right) - 1

    x = 3 \left(y^2 - \frac{5}{3}y + \frac{25}{36}\right) - 1 - \frac{25}{12}

    x = 3\left(y - \frac{5}{6}\right)^2 - \frac{37}{12}

    x + \frac{37}{12} = 3\left(y - \frac{5}{6}\right)^2

    \frac{x}{3} + \frac{37}{36} = \left(y - \frac{5}{6}\right)^2

    \sqrt{\frac{12x+37}{36}} = y - \frac{5}{6}

    \frac{5}{6} + \sqrt{\frac{12x+37}{36}} = y = f^{-1}(x)
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  3. #3
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    Re: Inverse function Help

    Skeeter "completed the square". You could also solve for y using the quadratic formula- which, of course, comes from "completing the square".
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  4. #4
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    Re: Inverse function Help

    When talking about a function, domain and range are necessary to be mentioned, a mere calculation without a domain does not qualify for a function. The original function is defined on [5/6, infinity) --> [-37/12, infinity), and the inverse is defined from [-37/12, infinity) --> [5/6, infinity). Making the function bijective like this is necessary for an inverse being defined.

    Salahuddin
    Maths online
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