tanhfunktiot

The tanh function, short for hyperbolic tangent, is a mathematical function with a distinctive S-shaped curve. It is defined as the ratio of the hyperbolic sine (sinh) to the hyperbolic cosine (cosh) of a variable, often denoted by x. Mathematically, this is expressed as tanh(x) = sinh(x) / cosh(x). Alternatively, it can be written in terms of the exponential function as tanh(x) = (e^x - e^-x) / (e^x + e^-x).

The domain of the tanh function is all real numbers, meaning it accepts any real number as input. However, its range is restricted to the open interval (-1, 1). This means that the output of the tanh function will always be a value strictly between -1 and 1, approaching -1 as the input approaches negative infinity and approaching 1 as the input approaches positive infinity. The function passes through the origin, with tanh(0) = 0.

The tanh function is an odd function, meaning that tanh(-x) = -tanh(x). Its derivative is given by 1 - tanh^2(x), which is always positive, indicating that the tanh function is strictly increasing. Due to its smooth, bounded, and zero-centered nature, the tanh function finds applications in various fields, particularly in machine learning as an activation function in artificial neural networks. It can help normalize outputs and introduce non-linearity into models.