Part 4/7:
Next, for hyperbolic cosine ( \cosh(x) = \frac{e^x + e^{-x}}{2} ), the graphical interpretation shows a distinctly different shape, being concave upwards. The inverse, ( \text{arc}\cosh(x) ), presents potential complications, as it can produce two outputs for any given ( x ). Thus, we impose constraints such that ( y \geq 0 ) to foster its one-to-one nature for functionality.
Hyperbolic Tangent
The hyperbolic tangent function, defined as ( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} ), follows a pattern that approaches asymptotes as ( y ) approaches either positive or negative infinity. The inverse ( \text{arc}\tanh(x) ) similarly finds resolution through reflection, producing an accurate depiction across specific quadrants of the Cartesian plane.