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Finding the antiderivative (inverse) of a function

  1. Jan 4, 2010 #1
    1. The problem statement, all variables and given/known data
    f(x)=1+e^x/1-e^x


    2. Relevant equations
    truth is, I don't even know how to approach this, i know i have to swap the variables but now I'm all confused because a friend told me to do this, in this particular case
    f(x)=[1+(e^f)-1^x]/[1-(e^f)-1^x]

    3. The attempt at a solution
    i don't knolw where this -1^x came from, i would give it a shot at how to resolve for x but I really don't know how to get it down from that position as an exponent that is, any hint or advice is truly welcomed.

    edit: forget the antiderivative term used in the subject i dont know why i confused those two terms, i just dont know how to delete it now =P
     
    Last edited: Jan 4, 2010
  2. jcsd
  3. Jan 4, 2010 #2

    vela

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    Try multiplying the top and bottom by [tex]e^{-x/2}[/tex].

    Or try u=[tex]e^x[/tex], then partial fractions.
     
    Last edited: Jan 4, 2010
  4. Jan 4, 2010 #3

    HallsofIvy

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    Frankly, it sounds to me like you are completely out of your depth. Are you sure you belong in this course? I am particularly concerned that you say titled this "finding the anti-derivative (inverse) of a function". Typically people learn about "inverse functions" in a course at least a semester or a year before the hear about "antiderivatives".
    "Finding the anti-derivative" and "finding the inverse" of a function are NOT the same thing at all.

    To find the inverse function to f(x)= (1+e^x)/(1-e^x) (Please, please, please, use parentheses! What you wrote was, correctly 1+ (e^x/1)- e^x but I am sure you did not mean that.) I would first write y= (1+e^x)/(1-e^x) (because "y" is simpler to write than "f(x)") and, as you said, "swap" x and y: x= (1+ e^y)/(1- e^y). Now solve for y. Multiply on both sides by 1- e^y to get x(1- e^y)= x- xe^y= 1+ e^y. Now add xe^y to both sides and subtract 1 from both sides to get (1+ x)e^y= x- 1. Divide both sides by x+ 1: e^y= (x-1)/(x+1). Finally, we can solve for y by doing the "inverse function" to the exponential, the natural logarithm: y= ln((x-1)/(x+1)).
     
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