## anonymous 4 years ago show that there is only one way to write f as the sum of an even function and an odd function

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1. anonymous

one way is $\frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2}$ oh only one way...

2. anonymous

usual idea is to imagine there are are two ways and then show they are the same. so maybe put $f(x)=e(x)+o(x)=\frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2}$ where $$o(x)$$ is odd and $$e(x)$$ is even and then show that $$e(x)=\frac{f(x)+f(-x)}{2}$$ and $$o(x)=\frac{f(x)-f(-x)}{2}$$

3. anonymous

yes, that should work let me know if it is not clear

4. anonymous

nice explanation, thanks!!

5. anonymous

yw