SOLUTION: A function f is called even if f(-x)=f(x) and it is called odd if f(-x)=-f(x). Show that every function f can be written as the sum of an even function and an odd function.

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Question 36004: A function f is called even if f(-x)=f(x) and it is called odd if f(-x)=-f(x). Show that every function f can be written as the sum of an even function and an odd function.
Answer by venugopalramana(3286) About Me  (Show Source):
You can put this solution on YOUR website!
AS REGARDS EVEN /ODD FUNCTIONS,WHERE FROM YOU GOT THAT QUESTION?THE
QUESTION HAS SOME AMBIGUITIES
LET US SEE THE SIMPLE CASE OF POLYNOMIALS
F(X)=X IS ODD FUNCTION SINCE F(-X)=-X=-F(X)
F(X)=X^2 IS EVEN FUNCTION SINCE F(-X)=X^2=+F(X)
SO IF
F(X)=X+X^2..IT IS NEITHER EVEN NOR ODD SINCE
F(-X)=-X+X^2= NEITHER F(X)...NOR...-F(X)
SUCH FUNCTIONS IN THIS CASE YOU MAY SPLIT AND SAY
F(X)=E(X)+O(X)...WHERE
E(X)=X^2...EVEN FUNCTION AND
O(X)=X...ODD FUNCTION.
BUT SUPPOSE
F(X)=SQRT(X)...THEN HOW DO YOU SPLIT IT AS A SUM?UNLESS YOU WRITE IT
AS POWER SERIES
FOR EX...F(X)=E^X..NEITHER EVEN NOR ODD..
F(-X)=E^(-X)=1/E^X
IF YOU WRITE A POWER SERIES FOR
E^X=1+X+X^2/2!+X^3/3!+.....YOU MAY DO IT..
BUT F(X)=SQRT(X)..AT F(-X)..IT IS NOT DEFINED ..IT IS IMAGINARY..
CHECK BACK AND TELL ME THE ANTECEDNTS OF THE QUESTION.