This one is very old and standard.. but good for those who dont know....
Prove that a function can be written as a sum of an even and odd function.
(Also: What is the condition for this to be written as above!)
9its domain must be symmetrical abt o
sry jus typed as soon as i saw
62yes priyam it is the domain... my wrong...
I just din read properly.. all i read was symmetic :D :P
1f(x) + f(-x) =0 for odd functions
and
f(x) - f(-x) =0 for even functions
1assume domain symmetrical about origin
let f(x) be the function
f(x)=f(x)
or 2f(x)=2f(x)
=>2f(x)=f(x)+f(x)+f(-x)-f(-x)
=>2f(x)=(f(x)+f(-x))+(f(x)-f(-x))
=> f(x)= (f(x)+f(-x))/2 + (f(x)-f(-x))/2
for the first function replacing x→ -x does not change the function
hence it is even
for the second one x→ -x gives f(-x)=-f(x) hence it is even
thus we can show f(x) as a sum two functions one even and one odd
