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

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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one way is \[\frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2}\] oh only one way...
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 \(o(x)=\frac{f(x)-f(-x)}{2}\)
yes, that should work let me know if it is not clear

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nice explanation, thanks!!
yw

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