SOLUTION: Suppose that f(x) is defined ∀x ∈ (−∞,∞). Show that f(x) = e(x) + o(x), where e is an even function and o is an odd function.

Algebra ->  Decimal-numbers -> SOLUTION: Suppose that f(x) is defined ∀x ∈ (−∞,∞). Show that f(x) = e(x) + o(x), where e is an even function and o is an odd function.       Log On


   

Question 1162395: Suppose that f(x) is defined ∀x ∈ (−∞,∞). Show that f(x) = e(x) + o(x), where e is an even function and o is an odd function.
Answer by ikleyn(52787) About Me  (Show Source):
You can put this solution on YOUR website!
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f(x) = %28f%28x%29+%2B+f%28-x%29%29%2F2 + %28f%28x%29+-+f%28-x%29%29%2F2.



It is the decomposition of the function f(x) into the sum of two functions



    e(x) = %28f%28x%29+%2B+f%28-x%29%29%2F2  and  o(x) = %28f%28x%29+-+f%28-x%29%29%2F2.



The function e(x) is an even function;  the function o(x) is the odd function.



So, the formula (*)  gives you the required presentation.


It is a standard statement from the Elementary function theory for beginner university Math students.