The hyperbolic tangent, denoted as tanh, is a mathematical function that describes the ratio of the hyperbolic sine to the hyperbolic cosine. It is defined as tanh(x) = sinh(x) / cosh(x). The function takes real numbers as input and outputs values between -1 and 1, making it useful in various applications, including neural networks and signal processing. The graph of the hyperbolic tangent function resembles an S-shape, approaching -1 as x approaches negative infinity and 1 as x approaches positive infinity. It is an odd function, meaning that tanh(-x) = -tanh(x). This property, along with its smooth curve, makes it valuable in modeling growth and decay processes.