Part 7/7:
( y = \tanh^{-1}(x) = \frac{1}{2} \ln\left(\frac{1+x}{1-x}\right) )
Domain: ( -1 < x < 1 ).
Conclusion
In summary, inverse hyperbolic functions are essential in the mathematical toolkit, facilitating complex calculations across various disciplines. Understanding their definitions, graphical representations, and derivations cements their application in solving equations and navigating calculus effectively. The functions provide a mirror to their non-inverted forms, granting insight into their one-to-one relationships and constraints.
This journey through inverse hyperbolic functions serves to reinforce their relevance and utility, paving the way for more advanced mathematical exploration in future studies.