arccoshcosh

arccoshcosh is a compact notation for the composition of the inverse hyperbolic cosine function with the hyperbolic cosine function, written as arccosh(cosh(x)). In this sense, it is the function obtained by applying cosh to an argument x and then applying arccosh to the result.

Definition and domain. The cosh function, cosh: R → [1, ∞), is even and strictly increasing on the nonnegative half-line. Its inverse on [1, ∞) is arccosh: [1, ∞) → [0, ∞). Therefore the composition arccosh(cosh(x)) is well defined for all real x, and its domain is all real numbers.

Range and primary identity. The composition arccosh(cosh(x)) evaluates to the nonnegative value |x| for every real x. This follows because cosh(x) = cosh(-x) and arccosh maps [1, ∞) back to [0, ∞) in a way that recovers the nonnegative input. Consequently, arccoshcosh(x) = |x|.

Properties. The function arccoshcosh has a nonnegative range [0, ∞) and is piecewise smooth, with a cusp at x = 0 (matching the cusp of the absolute value function). Its derivative is undefined at x = 0 and equals 1 for x > 0 and −1 for x < 0, mirroring the derivative of |x|.

Notes and usage. The term arccoshcosh is uncommon in standard references; it is typically described as the composition arccosh ∘ cosh. The result simplifies to the absolute value function, illustrating how principal branches of inverse hyperbolic functions interact with even, noninjective outer functions. See also arccosh, cosh, hyperbolic functions, and their inverses.