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==Even–odd decomposition==
EveryIf a real function ,has thea ___domain of whichthat is self-symmetric with respect to the origin, it may be uniquely decomposed as the sum of an even and an odd function, which are called respectively the '''even part''' and the '''odd part''' of the function;, ifand. are onedefined definesby
|equation<math display = {{NumBlk||<mathblock>f_\text{ eeven}(x) = \frac {f(x)+f(-x)}{2} ,</math> |{{EquationRef|Eq.3}}}}▼{{Equation box 1
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▲|equation = {{NumBlk||<math>f_\text{e}(x) = \frac {f(x)+f(-x)}{2}</math>|{{EquationRef|Eq.3}}}}
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and
: <math >f(x) display= block>f_\text{ eodd}(x) += f_\ textfrac { o}f(x) -f(-x)}{2}.</math> ▼
thenIt is straightforward to verify that <math>f_\text{ eeven}</math> is even, <math>f_\text{ oodd}</math> is odd, and <math>f=f_\text{even}+f_\text{odd}.</math>▼{{Equation box 1
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|equation = {{NumBlk||<math>f_\text{o}(x) = \frac {f(x)-f(-x)}{2}</math>|{{EquationRef|Eq.4}}}}
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▲then <math>f_\text{e}</math> is even, <math>f_\text{o}</math> is odd, and
▲: <math>f(x)=f_\text{e}(x) + f_\text{o}(x).</math>
This decomposition is unique since, if
Conversely, if
:<math>f(x)=g(x)+h(x),</math>
where {{mvar|g}} is even and {{mvar|h}} is odd, then <math>g=f_\text{e}</math> and <math>h=f_\text{o},</math> since
For example, the [[hyperbolic cosine]] and the [[hyperbolic sine]] may be regarded as the even and odd parts of the exponential function, as the first one is an even function, the second one is odd, and
:<math>e^x=\underbrace{\cosh (x)}_{f_\text{eeeven}(x)} + \underbrace{\sinh (x)}_{f_\text{oodd}(x)}</math>.
==Further algebraic properties==
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