Sqrttanhx

Sqrttanhx refers to the real-valued function defined by f(x) = sqrt(tanh x). It is the composition of the hyperbolic tangent with the square root, yielding a nonnegative output for real inputs where tanh x is nonnegative.

Domain and range. For real values, tanh x is nonnegative exactly when x ≥ 0, so the real-valued function is defined on [0, ∞). Since tanh x ∈ [0,1) for real x, the range of f is [0,1). The value at x = 0 is f(0) = sqrt(tanh 0) = 0, and as x grows, tanh x approaches 1 from below, so f(x) approaches 1 from below.

Basic properties. The function is strictly increasing on its domain because tanh x is increasing on [0, ∞) and the square root function is increasing on [0, ∞). It is differentiable on (0, ∞) with derivative f'(x) = sech^2(x) / (2 sqrt(tanh x)). The derivative blows up as x → 0+, reflecting a cusp at the left endpoint. Asymptotically, for small x, tanh x ~ x, so f(x) ~ sqrt(x); for large x, tanh x ~ 1 − 2 e^{-2x}, so f(x) ~ sqrt(1 − 2 e^{-2x}) ≈ 1 − e^{-2x}.

Extensions and notes. Outside the real domain, the expression can be extended to complex values by choosing a branch for the square root, but such extensions produce complex outputs for x < 0. Sqrttanhx is not a common standard function, but it serves as a simple example of a non-linear, bounded, monotone transform obtained by composing tanh with a square root.

Applications and references. It may be encountered in mathematical explorations of function compositions, numerical examples, or as a nonnegative, bounded activation-like transform in contexts where inputs are restricted to nonnegative real values.