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Inverse Function

  1. Jan 22, 2010 #1
    I got a problem for this question.

    Given the function y=f(x)=6x^3+6x+2. Find f^-1(x).

    Can anybody tell me how to solve this function?


    So far I got this:

    x=6y^3+6y+2

    x-2=6y^3+6y


    I just couldn't solve for y variable. So, please tell what I do wrong?

    thanks
     
  2. jcsd
  3. Jan 22, 2010 #2

    CompuChip

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    Science Advisor
    Homework Helper

    It doesn't look like you are doing anything wrong.
    Just note that it is a cubic polynomial. So although
    x-2 = 6y3+6y
    will definitely have solutions for y(x), they may be non-trivial to write down.

    However, a solution exists.
     
  4. Jan 22, 2010 #3
    You couldn't because it is not that much easy. I'm going to sort some steps that you should carefully follow to get x in terms of y:

    1- Define two new variables u and v, with uv = 2 and x=u-v. Introduce this into your function and simplify the result.

    2- Put v=2/u in the new function.

    3- Now you have a simple function which is quadratic in u^3. From there, I assume, you know how to solve a quadratic equation for t=u^3 after taking y to the right side and letting one side just be lonely with a zero!

    4- Now from 1,2 and 3 we have [tex]x=u-2/u = t^{1/3}-2t^{-1/3}[/tex] and this is what you want since t is a function of y.

    AB
     
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