Inverse hyperbolic functions and their graphs

Inverse Hyperbolic Functions

An inverse hyperbolic function is the inverse of the hyperbolic function.
The most common inverse hyperbolic functions, and their standard abbreviations, are: arcsinh (x) or inverse hyperbolic sine of x, arccosh (x) or inverse hyperbolic cosine of x and arctanh (x) or inverse hyperbolic tangent of x.

Just like trigonometric functions, inverse hyperbolic functions also have some specific properties such as the symmetry and reflection properties.
Symmetry property: The function arcsinh (x) is an odd function, i.e., arcsinh (-x) = - arcsinh (x).
Reflection property: The function arccosh (x) behaves like the function log x, it reflects about the line y = -x.

Each inverse hyperbolic function has a specific set of allowable inputs (or domain) and outputs (or range).
The domain of arcsinh (x) is all real numbers, and its range is also all real numbers.
The domain of arccosh (x) is -1 ≤ x ≤ 1, and its range is all numbers greater than or equal to 0.
The domain of arctanh (x) is -1 < x < 1, and its range is all real numbers.

The graph of arcsinh (x) is a curve that passes through the origin (0,0) and it has no restrictions, spreading across all real numbers both horizontally and vertically.
The graph of arccosh (x) starts from (1,0) and spreads to the right, resembling the graph of log x.
The graph of arctanh (x) starts from (-1,-∞) and ends at (1,+∞), increasing as it moves right.

These functions also have associated identities, for instance: arcsinh (x) = ln(x + √(x² + 1)), arccosh (x) = ln(x + √(x² - 1)) (when x ≥ 1), and arctanh (x) = 0.5*ln((1 + x) / (1 - x)) (for -1 < x < 1).

Mastery of the identities and of the properties of these functions can greatly simplify problem-solving in many areas such as calculus, differential equations, and statistics.
Be sure to practice solving integrals and derivatives involving these functions to get comfortable using them in such contexts.
Understanding these functions also opens the way to a deeper appreciation of more advanced topics like complex analysis, where they play pivotal roles.