inverse function

A function is a mathematical rule that connects two non-empty subsets of the real numbers, typically denoted as \( A \subseteq \mathbb{R} \) and \( B \subseteq \mathbb{R} \).

Given two functions \( f(x) \) and \( g(x) \), the composite function \( g \circ f \) is defined as the function obtained by first applying \( f \) to the input \( x \), and then applying \( g \) to the result \( f(x) \). Formally:
\[
g \circ f = g(f(x))
\]

An inverse function \( f^{-1} \) exists for a function \( f : X \to Y \) if and only if \( f \) is bijective. The inverse \( f^{-1} : Y \to X \) satisfies \( f^{-1}(f(x)) = x \) for all \( x \in X \) and \( f(f^{-1}(y)) = y \) for all \( y \in Y \).

The cotangent function \( f(x) = \cot(x) \) assigns to each angle \( x \), expressed in radians, its corresponding cotangent value.

\[
\cot(x) = \frac{\cos(x)}{\sin(x)}
\]

The arcsine function is the inverse of the sine function restricted to ([-\pi/2,\pi/2]), with domain ([-1,1]) and range ([-\pi/2,\pi/2]).

The arccotangent function is the inverse of the cotangent, defined for all real numbers and decreasing from π to zero.

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 hyperbolic tangent is a hyperbolic function defined as the ratio of the hyperbolic sine to the hyperbolic cosine, with values ranging between $−1$ and $1$.

The hyperbolic cotangent function is the ratio of the hyperbolic cosine to the hyperbolic sine, defined for all nonzero real numbers.

**Derivative of the inverse function** is the derivative of an invertible function expressed in terms of the derivative of the original function. It provides the rate of change of the inverse function where the original derivative is nonzero.