Use of the fact that every function is sum of an odd and an even function.

Question

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...?

Answer 1

The idea it can be useful to generalize your result:

The group $\mathbb{Z}/2$ acts on $\mathbb{R}$ in the following way:

for each $x\in \mathbb{R}$ you have that $\bar{0}x:=x$ while $\bar{1}x:=-x$

The action can be induced also on the set $V$ of the function from $\mathbb{R}$ to $\mathbb{R}$, that is a $\mathbb{R}-$ vector space , in the following way:

for each $f\in V $ $\bar{0}f:=f$ while

$\bar{1}f=f(-\cdot)$

So you can observe that $\mathbb{Z}/2$ can be represented on the vector space $V$.

We denote $(\mathbb{Z}/2)^*$ the group of character of $\mathbb{Z}/2$ where a character is a morphism from $\mathbb{Z}/2$ to $\mathbb{R}$.

We define the following eigenspace related to $\mathbb{Z}/2$:

for each $\chi\in (\mathbb{Z}/2)^*$

$L_\chi:=\{f\in V: g f=\chi(g) f \forall g\in \mathbb{Z}/2\}$

You can observe that there are only 2 possibile character related to $\mathbb{Z}/2$:

$\chi_ 0:=1$

$\chi_1(\bar{0}):=1$ and $\chi_1(\bar{1})=-1$

Now you can observe that

$L_{\chi_0}= \{f: f(x)=f(-x) \forall x\in \mathbb{R}\}$

$L_{\chi_1}= \{f: f(x)=-f(-x) \forall x\in \mathbb{R}\}$

So you have that $L_{\chi_0}$ is the space of even function while $L_{\chi_1}$ is the space of odd function.

You have that for each $f\in V$ then

$\frac{f+\bar{1}f}{2}\in L_{\chi_0}$ while $\frac{f-\bar{1}f}{2}\in L_{\chi_1}$ and

$f= \frac{f+\bar{1}f}{2}+ \frac{f-\bar{1}f}{2}$

so you have that

$V=L_{\chi_0}\oplus L_{\chi_1}$

Now you can generalize this result to a general set $X$ when a finite abelian group $G$ acts on it:

For each $f\in V$, where $V$ is the $\mathbb{K}$ -vector space of the function from $X$ to $\mathbb{K}$, with $char(\mathbb{K})\neg | o(G)$, you have that

$f_\chi:=\frac{1}{o(G)}\sum_{g\in G}\frac{1}{\chi(g)}g f\in L_{\chi}$ for each $\chi\in G^*$ and

$f=\sum_{\chi\in G^*} f_\chi$ so

$V=\oplus_{\chi\in G^*} L_\chi$

This is a useful result that often can be used in algebraic geometry to study some variety that can be viewed as quozient with respect to another variety and a finite abelian gruop $G$ acts on that variety.

Answer 2

When you are confronted with a new mathematical problem there are a few helpful rules to get ahead. One of them is: Search for the symmetries! Often a problem has in its formulation an obvious symmetry $x\leftrightarrow -x$. Such a symmetry could force the solutions to be symmetric as well.

Another example: One of the most important functions in analysis is ${\rm cis}:\ t\mapsto e^{it}$. It has a fantastic functional equation: ${\rm cis}(s+t)={\rm cis}(s)\cdot {\rm cis}(t)$ for arbitrary complex $s$, $t$. Unfortunately this function ${\rm cis}$ is complex-valued. But ${\rm cis}$ has an even part $t\mapsto\cos t$ and (up to the factor $i$) an odd part $t\mapsto\sin t$. These symmetric, resp., antisymmetric parts of ${\rm cis}$ have an immense stating in all of analysis and its applications. Similarly with the even and odd parts $\cosh$ and $\sinh$ of $x\mapsto e^x$.

I'm not sure whether it is important make a general theory of this for high school students.

Answer 3

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.