Question

show that there is only one way to write f as the sum of an even function and an odd function

Answer 1

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

Answer 2

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 \(o(x)=\frac{f(x)-f(-x)}{2}\)

Answer 3

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

Answer 4

nice explanation, thanks!!

Answer 5

yw