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nth-root functions are odd for odd positive n |
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If the ___domain of a real function is self-symmetric with respect to the origin, then the function can be uniquely decomposed as the sum of an even function and an odd function.
==Early history==
The concept of even and odd functions appears to date back to the early 18th century, with [[Leonard Euler]] playing a significant role in their formalization. Euler introduced the concepts of even and odd functions (using Latin terms ''pares'' and ''impares'') in his work ''Traiectoriarum Reciprocarum Solutio'' from 1727. Before Euler, however, [[Isaac Newton]] had already developed geometric means of deriving coefficients of power series when writing the ''Principia'' (1687), and included algebraic techniques in an early draft of his ''Quadrature of Curves,'' though he removed it before publication in 1706. It is also noteworthy that Newton didn't explicitly name or focus on the even-odd decomposition, his work with power series would have involved understanding properties related to even and odd powers.
==Definition and examples==
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===Multiplication and division===
* The [[multiplication|product]] and [[Division (mathematics)|quotient]] of two even functions is an even function.
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* The product
* The
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===Composition===
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* The composition of an even function and an odd function is even.
* The composition of any function with an even function is even (but not vice versa).
===Inverse function===
* If an odd function is [[inverse function|invertible]], then its inverse is also odd.
==Even–odd decomposition==
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