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Indicate whether sech(x) is invertible for [0, infinity) and explain why

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Homework Statement



Indicate whether sech(x) is invertible for [0, infinity) and explain why.


The Attempt at a Solution



Sooo... I know that:

sech(x) = 2 / ( ex + e-x )​

I know that to get the inverse equation I'd need to swap the y and the x... but I'm trying to show whether it's invertible so I don't know how much that would do me. I think I need to prove that the equation is monotonic, ie the derivative should be > 0 ). Hence, I took the derivative of the equation to be:

y' = - 2 (ex - e-x) / (ex + e-x)2

Is that right? If so, is that enough to answer the question?

Thanks!
 
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Answers and Replies

  • #2
SammyS
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You could show that sech(x) is not 1 to 1.

What are sech(1) and sech(-1) ?
 
  • #3
HallsofIvy
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But the problem said invertible on [0, infinity) so x= -1 is not relevant.
 
  • #4
SammyS
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DUH !!

Sorry, I missed the [0, infinity) !

It's right there in the title and in the text!
 

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