document.write( "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. \n" ); document.write( "
Algebra.Com's Answer #22360 by venugopalramana(3286)\"\" \"About 
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AS REGARDS EVEN /ODD FUNCTIONS,WHERE FROM YOU GOT THAT QUESTION?THE
\n" ); document.write( "QUESTION HAS SOME AMBIGUITIES
\n" ); document.write( "LET US SEE THE SIMPLE CASE OF POLYNOMIALS\r
\n" ); document.write( "\n" ); document.write( "F(X)=X IS ODD FUNCTION SINCE F(-X)=-X=-F(X)
\n" ); document.write( "F(X)=X^2 IS EVEN FUNCTION SINCE F(-X)=X^2=+F(X)
\n" ); document.write( "SO IF
\n" ); document.write( "F(X)=X+X^2..IT IS NEITHER EVEN NOR ODD SINCE
\n" ); document.write( "F(-X)=-X+X^2= NEITHER F(X)...NOR...-F(X)
\n" ); document.write( "SUCH FUNCTIONS IN THIS CASE YOU MAY SPLIT AND SAY
\n" ); document.write( "F(X)=E(X)+O(X)...WHERE
\n" ); document.write( "E(X)=X^2...EVEN FUNCTION AND
\n" ); document.write( "O(X)=X...ODD FUNCTION.
\n" ); document.write( "BUT SUPPOSE
\n" ); document.write( "F(X)=SQRT(X)...THEN HOW DO YOU SPLIT IT AS A SUM?UNLESS YOU WRITE IT
\n" ); document.write( "AS POWER SERIES
\n" ); document.write( "FOR EX...F(X)=E^X..NEITHER EVEN NOR ODD..
\n" ); document.write( "F(-X)=E^(-X)=1/E^X
\n" ); document.write( "IF YOU WRITE A POWER SERIES FOR
\n" ); document.write( "E^X=1+X+X^2/2!+X^3/3!+.....YOU MAY DO IT..
\n" ); document.write( "BUT F(X)=SQRT(X)..AT F(-X)..IT IS NOT DEFINED ..IT IS IMAGINARY..
\n" ); document.write( "CHECK BACK AND TELL ME THE ANTECEDNTS OF THE QUESTION. \n" ); document.write( "
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