Even and odd functions: Difference between revisions

Revision as of 17:55, 28 May 2024

Revision as of 18:02, 28 May 2024

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===Odd functions===

[[Image:Function-x3.svg|right|thumb|<math>f(x)=x^3</math> is an example of an odd function.]]

A real function {{math|''f''}} is '''odd''' if

Again, let ''f'' be a real-valued function of a real variable. Then ''f'' is '''odd''' if the following equation holds for all ''x'' such that ''x'' and −''x'' are in the ___domain of ''f'':<ref name=FunctionsAndGraphs/>{{rp|p. 72}}

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

or equivalently

{{Equation box 1

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

|indent =

for all {{math|''x''}} such that {{math|''x''}} and {{math|−''x''}} are in the ___domain of the function.<ref name=FunctionsAndGraphs/>{{rp|p. 72}}

|title=

|equation = {{NumBlk||<math>f(-x) = -f(x)</math>|{{EquationRef|Eq.2}}}}

|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 odd function has rotational symmetry with respect to the [[Origin (mathematics)|origin]], meaning that its graph remains unchanged after [[Rotation (mathematics)|rotation]] of 180 [[Degree (angle)|degree]]s about the origin.