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Concept
Inverse function
Summary
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
Let f be a function whose domain is the set X, and whose codomain is the set Y. Then f is invertible if there exists a function g from Y to X such that for all and for all .
If f is invertible, then there is exactly one function g satisfying this property. 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.
The function f is invertible if and only if it is bijective. This is because the condition for all implies that f is injective, and the condition for all implies that f is surjective.
The inverse function f −1 to f can be explicitly described as the function
Inverse element
Recall that if f is an invertible function with domain X and codomain Y, then
for every and for every .
Using the composition of functions, this statement can be rewritten to the following equations between functions:
and
where idX is the identity function on the set X; that is, the function that leaves its argument unchanged. In , 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.
Official source
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