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Finding the inverse of a function

  1. May 30, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the inverse function of [itex] f(x)= 2x + lnx [/itex]

    2. The attempt at a solution

    So usually i would start out with
    [itex] y= 2x+lnx [/itex] (finding x terms of y)
    and then switch x and y... but i can't figure out how to find x. doesn't seem to be possible to factorize this... are there other ways of doing this?
  2. jcsd
  3. May 30, 2012 #2
    I'm 99% certain this function's inverse isn't expressible in terms of elementary functions.
  4. May 30, 2012 #3

    Ray Vickson

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    Science Advisor
    Homework Helper

    The solution cannot be expressed in terms of "elementary" functions. It can be solved in terms of the so-called Lambert W-function: the solution of [itex] y = 2x + \ln(x)[/itex] is
    [tex] x = e^{y - W(2e^y)}, [/tex]
    where W(v) (Lambert W-function) is defined as the root of the equation [itex] z \:e^z = v[/itex] which is analytic at v = 0.

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