The inverse of a number, or a reciprocal, is 1 divided by the number; for example, the inverse of 8 is 1/8 and the inverse of 3/5 is 5/3.
The inverse of a function or a transformation is the function or transformation that the reverses the effect of the function or transformation. For example, the inverse of addition is subtraction, and of clockwise rotation is anticlockwise rotation. For a function f (x), the function g(x) such that f (a) = b implies g(b) = a is described as the inverse of f (x). In practice, the inverse function of f(x) is written f -1(x). For example, the inverse of f (x) = ax + b is f –1(x) = (x - b)/a, since f –1(ax + b) = (ax + b – b)/a = ax/a = x.
The inverse of an element of a set, or a number, with respect to a particular operation, is what has to be combined with the element or number in order to obtain that operation's identity element. In other words, the inverse of element a is the element b such that a*b = e, where * is an algebraic operation and e is the identity element relative to the operation * of the set of which a and b are members. For example, a is the identity element of real numbers relative to multiplication: hence if a.b = –1, a is the inverse of b, b the inverse of a, relative to the multiplication of real numbers. (Moreover, a and b are reciprocals in this case.)
Inverse of a proposition
For a proposition h → c (read h implies c), the proposition not-h → c is described as its inverse.
Inverse trigonometric function
For a function of the form y = sin x, the function of the form x = sin–1 y (read as "x is the angle whose sine is y ") is described as its inverse. This may also be written as x = arc sin y.