       Re: Inverse Functions

• To: mathgroup at smc.vnet.net
• Subject: [mg3208] Re: Inverse Functions
• From: whitic at rpi.edu (whitic)
• Date: Sat, 17 Feb 1996 14:16:33 -0500
• Organization: Rensselaer Polytechnic Institute, Troy NY, USA
• Sender: owner-wri-mathgroup at wolfram.com

```jcharko at microage-ll.awinc.com (Julian Charko) wrote:

>As a relatively inexperienced user of Mathematica, could someone inform me
>how to get a closed-form expression--in terms of elementary functions if
>possible--for the inverse of the function

>                           f(x) = x^x

>Since the function is one-to-one over its domain, it does have a
>mathematical inverse.

>Thank You,

>Julian P. Charko, P. Eng.

What domain are you considering for this function? If you are taking
the domain to be where the function is real valued, i.e. x>0, then
this function is not one-to-one. By setting the derivative to zero you
will find a stationary point at approx. x=.367879... and this is
actually a minimum since the second derivative is > 0.

If you are confining the function a the region where it is one-to-one,
e.g. x>.5, then the inverse function theorem only says there exists an
inverse, it doesn't say that it can be written in terms of elementary
functions, and in this case it can not be.

If you need to do this numerically on Mathematica, you could define a
function such as:
finv[f_,x1_,x2_] := FindRoot[ f-x^x==0, {x,x1,x2}]
or something like this, where x1 and x2 is the interval where you
expect to find x.

Good luck
Chris Whiting
whitic at rpi.edu

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```

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