I was looking at the wiki page http://en.wikipedia.org/wiki/Even_and_odd_functions#The_sum_of_even_and_odd_functions and it says that to prove an even function plus an odd function, we first have to re-write the f(x) as
$f(x)$ as $f(x)/2 + f(x)/2 + f(-x)/2 - f(-x)/2$
Why has f(x) been written like this? For example, where on earth did "+ f(-x)/2" and "- f(-x)/2" come from? I have my own idea as to where these functions have come from, and you can read my understanding of it below (and tell me if my thinking is correct or incorrect).
Secondly, I don't see why they have re-written the functions as being divided by two. Why have they done that? I feel that I could happily re-write the function f(x) as
$f(x)$ as $f(x)/ + f(x)/ + f(-x)/ - f(-x)/$ But would that be incorrect? If so, why?
My thoughts to my questions:
Is each function f(x) divided by 2 because f(x) is a function for all real numbers? If not, why is each function divided by 2 in this proof?
Secondly, why is the function of f(x) re-written in that form? Is it because:
an even function is $f(x)/2 = f(-x)/2$ which is $0 = f(x)/2 - f(-x)/2$
and an odd function is $f(x)/2 = -f(-x)/2$ which is $0 = f(x)/2 + f(-x)/2$
and these added together makes the 're-written f(x) function' $f(x)$ as $f(x)/2 + f(x)/2 + f(-x)/2 - f(-x)/2$ ?
If not, please do try to explain so I can understand this proof. Thanks