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Math Help - Find the Inverse Function

  1. #1
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    Find the Inverse Function

    Find the inverse of this function.

    y = x^3/4 + x - 1

    x = y^3/4 + y - 1

    x + 1 = y^3/4 + y

    x + 1 = (1/4)y[y^2 + 4]

    Where do I go from here?
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  2. #2
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    Re: Find the Inverse Function

    Multiply both sides by 4 and let y = s-t. We want to satisfy the following system of equations: s^3-t^3 = 4(x+1) and 3st = 4.

    Replacing the 4(x+1), and the coefficient for y, we have:

    s^3-t^3 = (s-t)^3+3st(s-t) = (s^3-3s^2t+3st^2-t^3) + (3s^2-3st^2)

    That is obviously true. So, let's solve the system of equations:

    s^3-t^3 = 4(x+1) and 3st = 4. Solving for s in the second equation, we can plug it into the first:

    \left(\dfrac{4}{3t}\right)^3-t^3 = 4(x+1)

    Simplifying, we get 27t^6 + 108(x+1)t^3-64 = 0. We can now solve for t^3 by plugging it into the quadratic equation:

    t^3 = \dfrac{-108(x+1) \pm \sqrt{11,664(x+1)^2+6,912}}{54}

    Take the cube root of both sides, and you have t. Plug that in for s = \dfrac{4}{3t}, and that gives you s. Then, y = s-t.

    (This is the general method for solving a cubic equation).
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  3. #3
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    Re: Find the Inverse Function

    I'm sorry but what is the inverse?
    What is f^(-1)?
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  4. #4
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    Re: Find the Inverse Function

    t = \sqrt[3]{\dfrac{-108(x+1) \pm \sqrt{11,664(x+1)^2+6,912}}{54}}

    So, s = \dfrac{4}{3\sqrt[3]{\dfrac{-108(x+1) \pm \sqrt{11,664(x+1)^2+6,912}}{54}}}

    Then the inverse function is:

    y = \dfrac{4}{3\sqrt[3]{\dfrac{-108(x+1) \pm \sqrt{11,664(x+1)^2+6,912}}{54}}} - \sqrt[3]{\dfrac{-108(x+1) \pm \sqrt{11,664(x+1)^2+6,912}}{54}}
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    Re: Find the Inverse Function

    I thank you for your reply but this question is from a precalculus textbook. The answer cannot be so complicated at this level of learning.

    I am to interchange x and y and then isolate y. Finally, I am to replace y with the inverse notation. After all of this, I am to evaluate the inverse by letting x be 1/2.
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  6. #6
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    Re: Find the Inverse Function

    Here is Wolfram Alpha's answer: here.

    It gives the complete answer. I only gave you part of the answer. In general, cubic equations are extremely messy. If that came from a precalc textbook, then my guess is that you should state that the inverse function is too messy to calculate.
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  7. #7
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    Re: Find the Inverse Function

    You are right. It is extremely messy. I need to disregard this question.
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