Difference between revisions of "Inverse"
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In [[mathematics]], if ƒ is a [[function]] from a set A to a set B, then an '''inverse''' [[function]] for ƒ is a function from B to A, with the property that a round trip (a [[composition]]) from A to B to A (or from B to A to B) returns each element of the initial set to itself. Thus, if an input x into the [[function]] ƒ produces an output y, then inputting y into the inverse function produces the output x, and vice versa. | In [[mathematics]], if ƒ is a [[function]] from a set A to a set B, then an '''inverse''' [[function]] for ƒ is a function from B to A, with the property that a round trip (a [[composition]]) from A to B to A (or from B to A to B) returns each element of the initial set to itself. Thus, if an input x into the [[function]] ƒ produces an output y, then inputting y into the inverse function produces the output x, and vice versa. | ||
| − | A [[function]] ƒ that has an inverse is called invertible; the inverse function is then [[uniquely]] determined by ƒ and is denoted by ƒ−1 (read f inverse, not to be confused with [ | + | A [[function]] ƒ that has an inverse is called invertible; the inverse function is then [[uniquely]] determined by ƒ and is denoted by ƒ−1 (read f inverse, not to be confused with [https://en.wikipedia.org/wiki/Exponentiation exponentiation]). |
*Definition | *Definition | ||
| − | Let ƒ be a [[function]] whose [[domain]] is the set X, and whose [ | + | Let ƒ be a [[function]] whose [[domain]] is the set X, and whose [https://en.wikipedia.org/wiki/Codomain codomain] is the set Y. Then, if it exists, the inverse of ƒ is the function ƒ−1 with domain Y and codomain X, with the property: |
[[File:Inverse.jpg]] | [[File:Inverse.jpg]] | ||
| − | Stated otherwise, a [[function]] is invertible if and only if its inverse [[relation]] is a [[function]], in which case the inverse relation is the inverse function.[ | + | Stated otherwise, a [[function]] is invertible if and only if its inverse [[relation]] is a [[function]], in which case the inverse relation is the inverse function.[https://en.wikipedia.org/wiki/Inverse_function] |
[[Category: Mathematics]] | [[Category: Mathematics]] | ||
Latest revision as of 01:42, 13 December 2020
In mathematics, if ƒ is a function from a set A to a set B, then an inverse function for ƒ is a function from B to A, with the property that a round trip (a composition) from A to B to A (or from B to A to B) returns each element of the initial set to itself. Thus, if an input x into the function ƒ produces an output y, then inputting y into the inverse function produces the output x, and vice versa.
A function ƒ that has an inverse is called invertible; the inverse function is then uniquely determined by ƒ and is denoted by ƒ−1 (read f inverse, not to be confused with exponentiation).
- Definition
Let ƒ be a function whose domain is the set X, and whose codomain is the set Y. Then, if it exists, the inverse of ƒ is the function ƒ−1 with domain Y and codomain X, with the property:
Stated otherwise, a function is invertible if and only if its inverse relation is a function, in which case the inverse relation is the inverse function.[1]
