I'm explaining to a student that all functions can be divided into odd and even (symmetric and anti-symmetric) components. It is easy to prove (basic algebra or Taylor series) but is not referenced in the text. I'm using it in relation to the definition of trigonometric functions.
Does this theorem have a name? Are there any good online resources for this that cover this at a High School/Calc I level (i.e. no Fourier series). Or is this considered simply too obvious to cover formally?
Take the function $f(x)$ and write the functions $\frac{1}{2}(f(x)-f(-x))$ and $\frac{1}{2}(f(x)+f(-x))$. They are odd and even, respectively, and their sum is $f(x)$. Nothing fancy required so I doubt it is named.
Edit: If you wanted to generalize this to the action of $n$th roots of unity as @KCd suggests, you could decompose the function $f(z)$ into functions $f_k(z)$ of the form $$ f_k(z) = \frac{1}{n} \sum_{j=0}^n \omega^{kj} f(\omega^j z) $$ where $\omega$ is a primitive $n$th root of unity. It is easy to check that $f= \sum_k f_k$ and that $f_k(\omega z) = \omega^k f(z)$. The odd/even case then corresponds to $n=2$, $\omega=-1$. You might even be able to explain that to advanced high school students if you can get them to understand why $\sum_{j=0}^n \omega^j = 0$. However, at that stage, the potential for confusion is probably not worth the reward.
