a. Even-odd decompositions Let ƒ be a function whose domain is symmetric about the origin, that is, -x belongs to the domain whenever x does. Show that ƒ is the sum of an even function and an odd function: ƒ(x) = E(x) + O(x), where E is an even function and O is an odd function. (Hint: Let E(x) = (ƒ(x) + ƒ(–x))/2. Show that E(–x) = E(x) so that E is even. Then show that O(x) = ƒ(x) – E(x), is odd.)
b. Uniqueness Show that there is only one way to write ƒ as the sum of an even and an odd function. (Hint: One way is given in part (a). If also ƒ(x) = E1(x) + O1(x) where E1is even and O1is odd, show that E - E1 = O1 - O.Then use
Exercise 11 to show that E = E1 and O = O1.
Reference: Exercise 11
Show that if ƒ is both even and odd, then ƒ(x) = 0for every x in the domain of ƒ.
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