What is inverse?

An inverse is a mathematical operation that undoes or reverses another operation. The inverse of addition is subtraction, the inverse of multiplication is division, the inverse of exponentiation is root extraction, and the inverse of trigonometric functions like sine and cosine are arcsine and arccosine, respectively.

Inverses play an important role in solving equations, particularly in calculus, linear algebra, and abstract algebra. Inverse functions are functions that, when composed with the original function, produce the identity function. For example, the inverse of the function f(x) = 2x + 3 is f⁻¹(x) = (x - 3)/2. The inverse function "undoes" the original function to give the original input.

Inverse matrices are another important concept in mathematics. A matrix has an inverse if and only if its determinant is non-zero. The inverse matrix can be used to solve systems of linear equations and to find the inverse transformation of geometric figures.

Lastly, inverses are also used in probability and statistics. The inverse cumulative distribution function gives the value at which a given probability is reached. The inverse of a probability density function is the cumulative distribution function.