Find the inverse of f(x) in 4 minutes

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Discussion Overview

The discussion revolves around finding the inverse of the function \( f(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} \). Participants explore the mathematical steps involved in deriving the inverse, as well as the context of a time-limited challenge.

Discussion Character

  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant questions the necessity of a time limit for the task.
  • Another participant identifies the function as the hyperbolic tangent, \( \tanh x \), and relates it to its inverse, \( \tanh^{-1} x \) or \( \artanh x \), providing a logarithmic expression for the inverse.
  • Several participants express light-heartedness about the time constraint and the challenge, suggesting a playful atmosphere.

Areas of Agreement / Disagreement

There appears to be a general agreement on the mathematical identity of the function and its inverse, but the discussion includes playful banter that does not contribute to a formal resolution of the time limit question.

Contextual Notes

The discussion does not delve into the implications of the time limit on the problem-solving process or the participants' approaches to the challenge.

Who May Find This Useful

Readers interested in hyperbolic functions, inverse functions, or those looking for a light-hearted mathematical challenge may find this discussion engaging.

Lorena_Santoro
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why a time limit?

$y=\dfrac{e^x-e^{-x}}{e^x+e^{-x}} = \dfrac{e^{2x}-1}{e^{2x}+1}$

$x = \dfrac{e^{2y}-1}{e^{2y}+1}$

$xe^{2y}+x = e^{2y}-1$

$x+1=e^{2y}(1-x)$

$e^{2y}=\dfrac{x+1}{1-x} \implies y = \dfrac{1}{2}\ln\left(\dfrac{x+1}{1-x}\right)$
 
It's a math-quiz like channel.
 
We might recognize the definition of $\sinh x=\frac 12(e^x-e^{-x})$ and its associates $\cosh x$ and $\tanh x$.
We have $\tanh x=\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}$, which is the same as the given $f(x)$.
Therefore $f^{-1}(x)=\tanh^{-1} x=\artanh x$, which happens to be the same as $\frac 12\ln\left(\frac{1+x}{1-x}\right)$ as skeeter showed. See Inverse hyperbolic tangent on wiki.
 
What should I do with the other three minutes?
 
Country Boy said:
What should I do with the other three minutes?
Enjoy being so smart! ;-)
 
And a nice cup of Earl Grey tea!
 
I'd totally agree!
 

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