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How to express a function as a function of another function?

  1. Feb 9, 2012 #1
    Hello,

    I would like to know how I could approach the following problem. I am given two functions [itex]y=f(x)[/itex] and [itex]z=g(x)[/itex], and I would like to express the first function as a function of the second one: that is, [tex]y = h(z)[/tex], where h is not necessarily a linear function of z.

    One explicit example could be: [tex]y=\frac{x}{a}[/tex] [tex]z=\frac{x}{a+b}[/tex]

    where the goal is to find a function h such that [tex]y=h(z)[/tex]
     
  2. jcsd
  3. Feb 9, 2012 #2

    tiny-tim

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    hello mnb96! :smile:

    invert g (if you can) …

    x = g-1(z)

    f(x) = f(g-1(z)) :wink:
     
  4. Feb 9, 2012 #3
    Ups...:)

    You are right. When the inverse for g exists, it is pretty easy. Thanks.
    I was wondering if it is still possible to do something when an inverse does not exist, although this goes slightly beyond the original question.
     
  5. Feb 9, 2012 #4

    tiny-tim

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    it'd have to be a pretty weird function not to have at least a local inverse :wink:
     
  6. Feb 10, 2012 #5
    Indeed... but while they may have a inverse, it may be hard (or even impossible) to write down this inverse function... unless "cheating" is allowed, like using functions like Maple's RootOf( f(x) ) ...
     
  7. Feb 11, 2012 #6

    tiny-tim

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    but we can still write it as g-1 :wink:
     
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