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August 6th, 2014, 08:04 AM  #1 
Newbie Joined: Aug 2013 Posts: 11 Thanks: 0  Is this function invertible?
Hello. How can one know if this Code: Exp[a Exp[x b]  c Exp[x d]] Code: Exp[a Exp[x/b]  c Exp[x/d]] Thank you in advance. Last edited by skipjack; August 6th, 2014 at 09:11 AM. 
August 6th, 2014, 08:12 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,681 Thanks: 2659 Math Focus: Mainly analysis and algebra 
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. 
August 6th, 2014, 09:15 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 20,966 Thanks: 2216 
There are some easy special cases, such as for c = 0.

August 6th, 2014, 06:46 PM  #4 
Math Team Joined: May 2013 From: The Astral plane Posts: 2,266 Thanks: 934 Math Focus: Wibbly wobbly timeywimey stuff. 
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 
August 7th, 2014, 01:02 AM  #5 
Newbie Joined: Aug 2013 Posts: 11 Thanks: 0 
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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