Real Function is Expressible as Sum of Even Function and Odd Function/Examples/Arbitrary Function 1

Example of Use of Real Function is Expressible as Sum of Even Function and Odd Function
Let $f: \R \to \R$ denote the real function:
 * $\map f x = e^{2 x} \sin x$

$f$ can be expressed as the pointwise sum of:
 * the even function $\map g x = \dfrac {\paren {e^{2 x} - e^{-2 x} } \sin x} 2$

and:
 * the odd function $\map h x = \dfrac {\paren {e^{2 x} + e^{-2 x} } \sin x} 2$

Proof
and:

The result follows.