functionsasinh

The hyperbolic sine inverse, often denoted as asinh(x) or sinh⁻¹(x), is the inverse function of the hyperbolic sine function, sinh(x). The hyperbolic sine function is defined as sinh(x) = (eˣ - e⁻ˣ) / 2. Because sinh(x) is a strictly increasing function, its inverse, asinh(x), is well-defined for all real numbers x.

The explicit formula for the hyperbolic sine inverse is given by asinh(x) = ln(x + √(x² + 1)). This formula can be derived by setting y = sinh(x) and solving for x. The domain of asinh(x) is all real numbers, (-∞, ∞), and its range is also all real numbers, (-∞, ∞).

The derivative of asinh(x) is 1 / √(x² + 1). This property makes it useful in calculus, particularly in integration. For instance, the integral of 1 / √(x² + 1) dx is asinh(x) + C, where C is the constant of integration.

The function asinh(x) is an odd function, meaning that asinh(-x) = -asinh(x). It is also continuous and differentiable for all real x. The graph of y = asinh(x) passes through the origin and has asymptotes as x approaches positive and negative infinity. It resembles the graph of the natural logarithm function but is steeper.