I'm not sure if it's a good question but I was reading about the decomposition of any function, f(x), as a sum of even and odd functions; f(x) is not an even or odd function.
Is it possible to know when this fact (or, theorem?) was discovered and who found it?
Helpful link: https://en.wikipedia.org/wiki/Even_and_odd_functions#Even%E2%80%93odd_decomposition
It depends partly on what "function" means. According to a definition prevalent before Dirichlet it meant something expressed by a formula and expandable into a power series. The decomposition claim then is obvious, as long as one introduced the notions of even and odd and related them to even and odd powers. Euler does both in Traiectoriarum Reciprocarum Solutio (1727), XIX-XXI (he uses Latin words pares and impares). Luckily, this is one of the papers translated by Ian Bruce:
"§17. In the first place functions are to be noted that I call even, of which this is the property: that they remain unchanged if $- x$ is put in place of $x$. All the powers of $x$ with even exponents are of this kind, and of fractions with even numerators and odd denominators. Then, any such functions of powers of this kind composed either by addition or subtraction, or by multiplication or division, or hence any such construction raised to sum are likewise even powers, such as $x^{\frac45}$, $(ax^2+bx^{\frac23})^n$.
§18. In the second place, I take heed of odd functions, which in short produce the negative of these, if $x$ is changed to $-x$. All the powers of x with odd exponents, such as $x$ itself, $x^3$, $x^5$, etc. are of this kind; or fractions of which the numerators and denominators are odd; also functions composed of these powers either added or subtracted, and which are also dignified to be raised to odd exponents, such as $x^{\frac45}$, $(ax^3+bx^{\frac57})^3$."
Euler did not have much use for them, or decomposition into them, although, as Barnett notes:
"In fact, the expressions $(e^x + e^{-x})/2$ and $(e^x - e^{-x})/2$ do make an appearance in Volume I of Euler's Introductio in analysin infinitorum (1745, 1748). Euler's interest in these expressions seems natural in view of the equations $\cos x=(e^{\sqrt{-1}x} + e^{-\sqrt{-1}x})/2$ and $\sqrt{-1}\sin x=(e^{\sqrt{-1}x} + e^{-\sqrt{-1}x})/2$ that he derived in this text. However, Euler's interest in what we call hyperbolic functions appears to have been limited to their role in deriving infinite product representations for the sine and cosine functions. Euler did not use the word hyperbolic in reference to the expressions... nor did he provide any special notation or name for them.
In a Hacker News discussion Decomposing a function into its even and odd parts David Goldblatt suggested that the idea might go back to earlier times:
"I don't have any references for you right now, but I would be surprised if the usage of the terms doesn't pre-date your citation of Euler by at least several decades. The basic technique for deriving Taylor series is known to date to at least 1671, to the correspondence between James Gregory and John Collins; it also shows up in letters from de Moivre to Johann Bernoulli in 1694 and from Leibniz to Bernoulli in 1708. Newton had a geometric means of deriving the coefficients of power series when he was writing the Principa (published in 1687), as demonstrated by the proof of his tenth proposition, and he included a description of the algebraic technique in an early draft of his Quadrature of Curves (but removed it before publication in 1706)... Perhaps nobody bothered to give the concept a name until Euler in 1727, and perhaps nobody even considered the concept with regard to a more general class of functions until even later".
