exponential function

A power shows how many times a number (the base) is multiplied by itself, or more generally, how it is raised to an exponent.

Hyperbolic sine and cosine describe the coordinates of a point on the equilateral hyperbola and are defined through symmetric combinations of exponential functions, providing a natural analogue to circular trigonometry.

The hyperbolic tangent and cotangent are functions defined as ratios of the hyperbolic sine and cosine, describing the relationship between these fundamental hyperbolic functions.

Exponential inequalities are inequalities in which the unknown appears in an exponent and are solved using common bases, logarithms, and monotonicity.

The exponential function \(y = a^x\) is a mathematical function in which a constant base \(a > 0\), \(a \neq 1\), is raised to the variable exponent \(x\). It describes exponential growth or decay and has domain \(\mathbb{R}\) and range \((0,\infty)\).

The hyperbolic sine is a hyperbolic function defined in terms of exponential functions, characterized by odd symmetry and closely related to the hyperbolic cosine.

The sigmoid function is a real-valued function of a real variable that takes values strictly between \(0\) and \(1\), approaching each of the two extremes asymptotically. It provides a smooth mapping from the real line to the unit interval and is widely used in analysis and machine learning. Its definition is the following:

\[
\sigma(x) = \frac{1}{1 + e^{-x}}
\]