Even and odd functions: Difference between revisions

Revision as of 18:23, 28 May 2024

Revision as of 07:45, 29 May 2024

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===Even functions===

[[Image:Function x^2.svg|right|thumb|<math>f(x)=x^2</math> is an example of an even function.]]

forA all[[real function]] {{math|''xf''}} suchis that'''even''' if, for every {{mathmvar|''x''}} andin its ___domain, {{math|−''x''}} areis also in its ___domain of the function.and<ref name=FunctionsAndGraphs>{{cite book|first1=I. M.|last1=Gel'Fand|author1-link=Israel Gelfand|first2=E. G.|last2=Glagoleva|author2-link=E. G. Glagoleva|first3=E. E.|last3=Shnol|title=Functions and Graphs|year=1990|publisher=Birkhäuser|isbn=0-8176-3532-7|url-access=registration|url=https://archive.org/details/functionsgraphs0000gelf}}</ref>{{rp|p. 11}}

A [[real function]] {{math|''f''}} is '''even''' if

<math display=block>f(-x) = f(x)</math>

or equivalently

<math display=block>f(x) - f(-x) = 0.</math>

for all {{math|''x''}} such that {{math|''x''}} and {{math|−''x''}} are in ___domain of the function.<ref name=FunctionsAndGraphs>{{cite book|first1=I. M.|last1=Gel'Fand|author1-link=Israel Gelfand|first2=E. G.|last2=Glagoleva|author2-link=E. G. Glagoleva|first3=E. E.|last3=Shnol|title=Functions and Graphs|year=1990|publisher=Birkhäuser|isbn=0-8176-3532-7|url-access=registration|url=https://archive.org/details/functionsgraphs0000gelf}}</ref>{{rp|p. 11}}

Geometrically, the graph of an even function is [[Symmetry|symmetric]] with respect to the ''y''-axis, meaning that its graph remains unchanged after [[Reflection (mathematics)|reflection]] about the ''y''-axis.