Not to be confused with multiplicative inverse or additive inverse.
A function f and its inverse f −1. Because f maps a to 3, the inverse f −1 maps 3 back to a.
Function
x ↦ f (x)
History of the function concept
Examples of domains and codomains
→,→,→
→,→
→,→,→
→,→,→
Classes/properties
Constant
Identity
Linear
Polynomial
Rational
Algebraic
Analytic
Smooth
Continuous
Measurable
Injective
Surjective
Bijective
Constructions
Restriction
Composition
λ
Inverse
Generalizations
Binary relation
Partial
Multivalued
Implicit
Space
v
t
e
In mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective, and if it exists, is denoted by
For a function , its inverse admits an explicit description: it sends each element to the unique element such that f(x) = y.
As an example, consider the real-valued function of a real variable given by f(x) = 5x − 7. One can think of f as the function which multiplies its input by 5 then subtracts 7 from the result. To undo this, one adds 7 to the input, then divides the result by 5. Therefore, the inverse of f is the function defined by
mathematics, the inversefunction of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if...
specifically differential calculus, the inversefunction theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point...
the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions...
mathematics, the inverse hyperbolic functions are inverses of the hyperbolic functions, analogous to the inverse circular functions. There are six in...
The reciprocal function, the function f(x) that maps x to 1/x, is one of the simplest examples of a function which is its own inverse (an involution)...
In mathematics, the inverse gamma function Γ − 1 ( x ) {\displaystyle \Gamma ^{-1}(x)} is the inversefunction of the gamma function. In other words, y...
calculus, the inversefunction rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of...
percentile function (after the percentile), percent-point function, inverse cumulative distribution function (after the cumulative distribution function) or...
In economics, an inverse demand function is the mathematical relationship that expresses price as a function of quantity demanded (it is therefore also...
considered above grows very rapidly, its inversefunction, f−1, grows very slowly. This inverse Ackermann function f−1 is usually denoted by α. In fact,...
mathematics, integrals of inversefunctions can be computed by means of a formula that expresses the antiderivatives of the inverse f − 1 {\displaystyle f^{-1}}...
from any probability distribution given its cumulative distribution function. Inverse transformation sampling takes uniform samples of a number u {\displaystyle...
\end{aligned}}} The inverse of Φ is known as the normal quantile function, or probit function and may be expressed in terms of the inverse error function as probit...
implicit function is an inversefunction. Not all functions have a unique inversefunction. If g is a function of x that has a unique inverse, then the...
follows that the composition of two bijections is also a bijection. The inversefunction of a composition (assumed invertible) has the property that (f ∘ g)−1...
above). Addition of real- and complex-valued functions: here, the additive inverse of a function f is the function −f defined by (−f )(x) = − f (x), for all...
the implicit function theorem. Inversefunction theorem Constant rank theorem: Both the implicit function theorem and the inversefunction theorem can...
inverse of a function, not exponentiation.) As sine and cosine are not injective, their inverses are not exact inversefunctions, but partial inverse...
trigonometric functions has a corresponding inversefunction, and an analog among the hyperbolic functions. The oldest definitions of trigonometric functions, related...
exponential function is a bijection from R {\displaystyle \mathbb {R} } to the interval ( 0 , ∞ ) {\displaystyle (0,\infty )} . Its inversefunction is the...
integers to the even numbers, which has the division by two as its inversefunction. A function is bijective if and only if it is both injective (or one-to-one)—meaning...
In mathematics, the logarithm is the inversefunction to exponentiation. That means that the logarithm of a number x to the base b is the exponent to which...
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating...
for the quotient rule. Suppose that y = g(x) has an inversefunction. Call its inversefunction f so that we have x = f(y). There is a formula for the...
More generally, a function has a left inverse for function composition if and only if it is injective, and it has a right inverse if and only if it is...
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