Find the inverse of f(x) in 4 minutes

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SUMMARY

The inverse of the function \( f(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} \) is \( f^{-1}(x) = \tanh^{-1}(x) = \frac{1}{2} \ln\left(\frac{1+x}{1-x}\right) \). This relationship is established through the manipulation of hyperbolic functions, specifically recognizing that \( f(x) \) is equivalent to \( \tanh(x) \). The discussion emphasizes the efficiency of deriving the inverse within a four-minute time frame, showcasing the mathematical elegance of hyperbolic identities.

PREREQUISITES
  • Understanding of hyperbolic functions, specifically \( \sinh \), \( \cosh \), and \( \tanh \)
  • Familiarity with logarithmic identities and properties
  • Basic algebraic manipulation skills
  • Knowledge of inverse functions and their derivations
NEXT STEPS
  • Study the properties and applications of hyperbolic functions in calculus
  • Learn about the derivation and applications of inverse hyperbolic functions
  • Explore the relationship between exponential functions and logarithms
  • Investigate the use of hyperbolic functions in solving differential equations
USEFUL FOR

Mathematicians, students studying calculus or advanced algebra, and anyone interested in the applications of hyperbolic functions in mathematical analysis.

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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