Use of the fact that every function is sum of an odd and an even function. It is well know that every real variable function $f$ can be written as a sum of an odd and an even function, namely $h$ and $g$ where: $$h(x) = {f(x)-f(-x)\over 2}\;\;\;\;\;\;\;\;\;\;\;\;g(x) = {f(x)+f(-x)\over 2}$$
Now what is the use of this fact? I told that to my $\color{red}{\rm high\; school}$ students but then I don't know what to do with this fact. Is it good for a graphing or to calculate the zeroes of a function or something else...?

You could tie it to Fourier series. Not formally of course but just hint at the really neat idea that you can write a function as a series of sin (odd) and cos (even) functions.
You could also tie it to Taylor series (again not formally just a little introduction) where functions are the sums of odd powers (odd functions) and even powers (even functions). And, you could show in your examples that odd functions like sin(x) are the sums of odd powers in the Taylor series.
For actual applications, I personally have never had to use that fact to solve a problem, so I wouldn't worry about its application as a tool in High School math.
