Even and odd functions: Difference between revisions

Revision as of 17:18, 28 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.]]

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

Let ''f'' be a real-valued function of a real variable. Then ''f'' is '''even''' if the following equation holds for all ''x'' such that ''x'' and −''x'' are in the ___domain of ''f'':<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}}

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

or equivalently

{{Equation box 1

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

|indent =

Let ''f'' be a real-valued function of a real variable. Then ''f'' is '''even''' if the following equation holds for all {{math|''x''}} such that {{math|''x''}} and {{math|−''x''}} are in the ___domain of ''f'':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}}

|title=

|equation = {{NumBlk||<math>f(-x) = f(x)</math>|{{EquationRef|Eq.1}}}}

|cellpadding= 6

|border

|border colour = #0073CF

|background colour=#F5FFFA}}

or equivalently if the following equation holds for all such ''x'':

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

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.