Part 5/7:
Deriving the Inverse Functions
The derivations of these inverse functions follow a common pathway. Let's summarize important formulations:
Inverse Hyperbolic Sine
Starting with the equation ( x = \sinh(y) ):
Rearranging gives us ( x = \frac{e^y - e^{-y}}{2} ).
Clearing the fractions and organizing leads us to a quadratic equation.
Solving this quadratic grants us ( y = \ln(x + \sqrt{x^2 + 1}) ).
This entire process highlights the transformation of the original function into its respective inverse.
Inverse Hyperbolic Cosine
Following a similar method, we start at ( x = \cosh(y) ):
- Manipulating the expression ( x = \frac{e^y + e^{-y}}{2} ) enables another quadratic formation.