Identity function - wikipedia Identity function Not to be confused with Null function or Empty function. Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument. In equations, the function is given by f ( x ) = x. Contents 1 Definition 2 Algebraic property 3 Properties 4 See also 5 References Definition Formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfies f ( x ) = x for all elements x in M. In other words, the function value f ( x ) in M ( that is, the codomain ) is always the same input element x of M ( now considered as the domain ). The identity function on M is clearly an injective function as well as a surjective function, so it is also bijective. The identity function f on M is often denoted by id. In set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or diagonal of M. Algebraic property If f : M → N is any function, then we have f ∘ id = f = id ∘ f ( where `` ∘ '' denotes function composition ). In particular, id is the identity element of the monoid of all functions from M to M. Since the identity element of a monoid is unique, one can alternately define the identity function on M to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of M need not be functions. Properties The identity function is a linear operator, when applied to vector spaces. The identity function on the positive integers is a completely multiplicative function ( essentially multiplication by 1 ), considered in number theory. In an n - dimensional vector space the identity function is represented by the identity matrix I, regardless of the basis. In a metric space the identity is trivially an isometry. An object without any symmetry has as symmetry group the trivial group only containing this isometry ( symmetry type C ). In a topological space, the identity function is always continuous. See also Inclusion map References Jump up ^ Knapp, Anthony W. ( 2006 ), Basic algebra, Springer, ISBN 978 - 0 - 8176 - 3248 - 9 Jump up ^ Mapa, Sadhan Kumar. Higher Algebra Abstract and Linear ( 11th ed. ). Sarat Book House. p. 36. ISBN 978 - 93 - 80663 - 24 - 1. Jump up ^ Anton, Howard ( 2005 ), Elementary Linear Algebra ( Applications Version ) ( 9th ed. ), Wiley International Jump up ^ D. Marshall ; E. Odell ; M. Starbird ( 2007 ). Number Theory through Inquiry. Mathematical Association of America Textbooks. Mathematical Assn of Amer. ISBN 978 - 0883857519. Jump up ^ T.S. Shores ( 2007 ). Applied Linear Algebra and Matrix Analysis. Undergraduate Texts in Mathematics. Springer. ISBN 038 - 733 - 195 - 6. Jump up ^ James W. Anderson, Hyperbolic Geometry, Springer 2005, ISBN 1 - 85233 - 934 - 9 Retrieved from `` https://en.wikipedia.org/w/index.php?title=Identity_function&oldid=790102911 '' Categories : Functions and mappings Elementary mathematics Basic concepts in set theory Types of functions 1 ( number ) Talk Contents About Wikipedia Bosanski Català Corsu Dansk Deutsch Español Euskara فارسی Français Galego 한국어 हिन्दी Íslenska Italiano עברית Lumbaart Nederlands नेपाली 日本 語 Polski Português Русский Slovenščina Suomi Svenska தமிழ் ไทย Türkçe Українська Tiếng Việt 中文 Edit links This page was last edited on 11 July 2017, at 16 : 44. Text is available under the Creative Commons Attribution - ShareAlike License ; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia ® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. About Wikipedia

Question: f x x is what type of function
Answer: an identity function, also called an identity relation or identity map or identity transformation