Question

How to write a function that expresses f(x)

Answer 1

lets write

(x) = (1 / 2) f(x) + (1 / 2) f(x) + (1 / 2) f(-x) - (1 / 2) f(-x)

which gives us

= (1 / 2) (f(x) + f(-x)) + (1 / 2)(f(x) - f(-x))

let

g(x) = (1 / 2) (f(x) + f(-x))

check that g(x) is even

g(-x) = (1 / 2) (f(-x) + f(x)) = g(x)

let

h(x) = (1 / 2) (f(x) - f(-x))

check that h(x) is odd

h(-x) = (1 / 2) (f(-x) - f(x)) = - (1 / 2) (f(x) - f(-x)) = - h(x)

Explanation:

Answer 2

lets write
(x) = (1 / 2) f(x) + (1 / 2) f(x) + (1 / 2) f(-x) - (1 / 2) f(-x)
which gives us
= (1 / 2) (f(x) + f(-x)) + (1 / 2)(f(x) - f(-x))
let
g(x) = (1 / 2) (f(x) + f(-x))
check that g(x) is even
g(-x) = (1 / 2) (f(-x) + f(x)) = g(x)
let
h(x) = (1 / 2) (f(x) - f(-x))
check that h(x) is odd
h(-x) = (1 / 2) (f(-x) - f(x)) = - (1 / 2) (f(x) - f(-x)) = - h(x)