Even and odd functions: Difference between revisions

Revision as of 11:26, 22 January 2024

Revision as of 02:15, 23 January 2024

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{{distinguish|Even and odd numbers}}

[[File:Sintay SVG.svg|thumb|The [[sine function]] and all of its [[Taylor polynomial]]s are odd functions.]]

[[File:Développement limité du cosinus.svg|thumb|The [[cosine function]] and all of its [[Taylor polynomials]] are even functions.∮]]

In [[mathematics]], an '''even function''' is a [[functionreal (mathematics)|function]] such that <math>f(-x)=f(x)</math> for every <math>x</math> in its [[___domain of a function|___domain]]. Similarly, an '''odd function''' is a function such that <math>f(-x)=-f(x)</math> for every <math>x</math> in its ___domain.

They are named for the [[parity (mathematics)|parity]] of the powers of the [[Power Function|power functions]] which satisfy each condition: the function <math>f(x) = x^n</math> is even if ''n'' is an [[even integer]], and it is odd if ''n'' is an odd integer.

Even functions are those real functions whose [[graph of a function|graph]] is [[symmetry (geometry)|self-symmetric]] with respect to the {{nowrap|{{mvar|y}}-axis,}} and odd functions are those whose graph is self-symmetric with respect to the [[origin (mathematics)|origin]].

If the ___domain of a real function is self-symmetric with respect to the originm then the function can be uniquely decomposed as the sum of an even function and an odd function.

==Definition and examples==