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August 6th, 2014, 08:04 AM   #1
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Is this function invertible?

Hello.
How can one know if this
Code:
Exp[-a Exp[-x b] - c Exp[-x d]]
function, or this similar other
Code:
Exp[-a Exp[-x/b] - c Exp[-x/d]]
are invertible?. Or maybe they have a lot of possible solutions?

Thank you in advance.

Last edited by skipjack; August 6th, 2014 at 09:11 AM.
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August 6th, 2014, 08:12 AM   #2
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I suspect that they have a global minimum (less than zero) somewhere in $x \lt 0$, so they probably don't have an inverse defined on the whole of the range. You could confirm/deny this by examining the derivative.

Such inverses as they do have are unlikely to have a finite representation in standard functions.

Last edited by v8archie; August 6th, 2014 at 08:14 AM.
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August 6th, 2014, 09:15 AM   #3
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There are some easy special cases, such as for c = 0.
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August 6th, 2014, 06:46 PM   #4
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Assuming c is not 0, here is one possibility: a = c = 1, b = 2, d = 1.

As you can see the function is not 1 - 1, which is a requirement for the existence of an inverse function.

-Dan
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August 7th, 2014, 01:02 AM   #5
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Thank you very much for your posts.

I want to clarify one thing: we suppose the constants a,b,c and d are higher than 0.

And I want to ask other regarding v8archie response: Are there any rules to confirm whether the function is invertible or not? Could you give me any reference?

Thanks.

Last edited by skipjack; August 10th, 2014 at 11:08 AM.
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