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Re: finding inverses of functions

• To: mathgroup at smc.vnet.net
• Subject: [mg126442] Re: finding inverses of functions
• From: Andrzej Kozlowski <akozlowski at gmail.com>
• Date: Thu, 10 May 2012 05:01:13 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201205090750.DAA05576@smc.vnet.net>

On 9 May 2012, at 09:50, John Accardi wrote:

> Hello,  I am trying to get Mathematica to find the inverse of:
>
> y = 3x^3 + 2e^(2x)  (which I know is invertible)
>
> InverseFunction only seems to give inverses of built-ins, like Sine.
>
> I tried:
>
> Solve[ y == 3x^3 + 2e^(2x), x ] but get a message that Solve does 
not have methods suitable.  (Solve works for simpler functions, 
however.)
>
> Any ideas?  Thanks.

First, you have e instead of E in your definition which makes nonsense 
of the entire problem as far as Mathematica is concerned.

Secondly, what do you mean by "finding the inverse"? Your function is 
clearly invertible (only as a function on the real line, of course) but 
there is no way to express this inverse in terms of known functions. 
However, you can define the inverse yourself:

ff[y_] :=
 Block[{x}, x /. NSolve[y == 3 x^3 + 2 E^(2 x), x, Reals][[1]]]

To see that this works, define:

gg[x_] := 3 x^3 + 2 E^(2 x)

and look at the graphs:

DiscretePlot[ff[gg[x]], {x, -1, 1, 1/10}]

DiscretePlot[gg[ff[x]], {x, -1, 1, 1/10}]


Andrzej Kozlowski

• References:
  • finding inverses of functions
    • From: John Accardi <accardi@accardi.com>