Every [[Function]] may be uniquely **decomposed** as the sum of - an [[even function]] (the *even part*) $f_{\mathrm{e}}(x)=\frac{f(x)+f(-x)}{2}$ and - an [[odd function]] (the *odd part*) $f_{\mathrm{o}}(x)=\frac{f(x)-f(-x)}{2}$ Leading to the decomposition $f(x)=f_{\mathrm{e}}(x)+f_{\mathrm{o}}(x)$ ##### Examples: Decompose [[𝑒^x - Exponential Function]] into [[hyperbolic cosine function]] and [[hyperbolic sine function]] $e^{x}=\underbrace{\cosh (x)}_{f_{\mathrm{e}}(x)}+\underbrace{\sinh (x)}_{f_{\mathrm{o}}(x)}$