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MathGroup Archive 2007

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Re: How to invert a function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg81179] Re: How to invert a function
  • From: Bill Rowe <readnewsciv at sbcglobal.net>
  • Date: Fri, 14 Sep 2007 03:47:55 -0400 (EDT)

On 9/13/07 at 6:24 AM, jwheeler51 at gmail.com (Teodoro) wrote:

>Hi, this is my first post ro the Mathematica group, so, please be
>patient ... The problem I'm trying to solve is a little bit more
>complex, but anyway I was able to reproduce the unwanted behaviour
>using a "toy" example. As far as I understand the fragment of code

>m = 7.984136668700428`*^-14; p = 38.64734299516908`; n =
>9.185777`*^-7; q = 7.729468599033817`; r = -9.18579746734159`*^-7;
>f[y_] = m Exp[p y];
>g[x_] := Solve[f[y] == x, y]
>h[x_] := y /. g[x]

>allows me to get y as a function of x. To check that h[x] is indeed
>the inverse of f[y] (you already know the solution !) one can run

>Evaluate[f[h[z]]]
>Evaluate[h[f[z]]]

>for any value of z you get again z:

>z -> h[z] -> f[h[z]=z
>z -> f[z] -> h[f[z]=z

>or

>Plot[Evaluate[h[x]], {x, 1, 20}, PlotRange -> {Full}]

>and get a straight line. However, when I try to invert

>f[y_] = m Exp[p y]+n Exp[q y]+r

>I get some result, but the result is WRONG !

This really shouldn't be surprising. What really should be
surprising is that you get anything other than error messages.
Consider what you are asking keeping in mind Solve is designed
to give symbolic answers not numeric answers.

=46or your first f you have m Exp[p y] which is easily inverted
using Log, i.e.,

In[26]:= Solve[m Exp[p y] == x, y]

Out[26]= {{y -> Log[x/m]/p}}

works and simply gives you a warning about using inverse functions

Now try

Solve[m Exp[p y] + n Exp[q y] + r == x, y]

and you will get an error message saying in essence there is no
symbolic solution for y possible. That is you need to use
numeric techniques here

There are a number of functions available in Mathematica to do
this, such as NSolve or FindRoot. Note FindRoot would be the
better choice for this function.
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