Inverse function
admits an explicit description: it sends each element
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.
The function g is called the inverse of f, and is usually denoted as f −1, a notation introduced by John Frederick William Herschel in 1813.
In category theory, this statement is used as the definition of an inverse morphism.
Considering function composition helps to understand the notation f −1.
Repeatedly composing a function f: X→X with itself is called iteration.
If f is applied n times, starting with the value x, then this is written as f n(x); so f 2(x) = f (f (x)), etc.
Since f −1(f (x)) = x, composing f −1 and f n yields f n−1, "undoing" the effect of one application of f. While the notation f −1(x) might be misunderstood,[1] (f(x))−1 certainly denotes the multiplicative inverse of f(x) and has nothing to do with the inverse function of f.[6] The notation
[7] In keeping with the general notation, some English authors use expressions like sin−1(x) to denote the inverse of the sine function applied to x (actually a partial inverse; see below).
[6] To avoid any confusion, an inverse trigonometric function is often indicated by the prefix "arc" (for Latin arcus).
[9][10] Similarly, the inverse of a hyperbolic function is indicated by the prefix "ar" (for Latin ārea).
[10] For instance, the inverse of the hyperbolic sine function is typically written as arsinh(x).
has an explicit description as This allows one to easily determine inverses of many functions that are given by algebraic formulas.
The formula for this inverse has an expression as an infinite sum: Since a function is a special type of binary relation, many of the properties of an inverse function correspond to properties of converse relations.
) is invertible on its range (image) if and only if it is either strictly increasing or decreasing (with no local maxima or minima).
Specifically, a continuously differentiable multivariable function f : Rn → Rn is invertible in a neighborhood of a point p as long as the Jacobian matrix of f at p is invertible.
In this case, the Jacobian of f −1 at f(p) is the matrix inverse of the Jacobian of f at p. Even if a function f is not one-to-one, it may be possible to define a partial inverse of f by restricting the domain.
The most important branch of a multivalued function (e.g. the positive square root) is called the principal branch, and its value at y is called the principal value of f −1(y).
For a continuous function on the real line, one branch is required between each pair of local extrema.
The above considerations are particularly important for defining the inverses of trigonometric functions.
However, the sine is one-to-one on the interval [−π/2, π/2], and the corresponding partial inverse is called the arcsine.
A function f with nonempty domain is injective if and only if it has a left inverse.
[21] An elementary proof runs as follows: If nonempty f: X → Y is injective, construct a left inverse g: Y → X as follows: for all y ∈ Y, if y is in the image of f, then there exists x ∈ X such that f(x) = y.
Otherwise, let g(y) be an arbitrary element of X.For all x ∈ X, f(x) is in the image of f. By construction, g(f(x)) = x, the condition for a left inverse.In classical mathematics, every injective function f with a nonempty domain necessarily has a left inverse; however, this may fail in constructive mathematics.
For instance, a left inverse of the inclusion {0,1} → R of the two-element set in the reals violates indecomposability by giving a retraction of the real line to the set {0,1}.
If f: X → Y is any function (not necessarily invertible), the preimage (or inverse image) of an element y ∈ Y is defined to be the set of all elements of X that map to y: The preimage of y can be thought of as the image of y under the (multivalued) full inverse of the function f. The notion can be generalized to subsets of the range.
This function is not invertible as it is not bijective, but preimages may be defined for subsets of the codomain, e.g.
The original notion and its generalization are related by the identity
The preimage of a single element y ∈ Y – a singleton set {y} – is sometimes called the fiber of y.
