Proof Involving Functions g & h

a. given a function f, prove that g is even and h is odd, where
g(x)=(1/2)(f(x)+f(-x))
and
h(x)=(1/2)(f(x)-f(-x))

to prove that g is even, show that g(-x)=g(x)

g(-x)=(1/2)(f(-x)+f(-(-x)))
g(-x)=(1/2)(f(-x)+f(x))) ....use comutation
g(-x)=(1/2)(f(x)+f(-x)))
g(-x)=g(x)->proven

Check that h(x) is odd, or if h(-x)= - h(x)

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



b.
use the result of part (a) to prove that any function can be written as a sum of even and odd functions.
f(x) =g(x)+h(x)
f(x) =(1/2)(f(x)+f(-x)) +(1/2)(f(x)-f(-x))
f(x) =(1/2)(f(x)+f(-x) +f(x)-f(-x))
f(x) =(1/2)(2f(x))
f(x) =f(x)-> proven that f(x) =g(x)+h(x)

c.
use the result of part (b) to write each function as a sum of even and odd functions


f(x) =x^2-2x+1
f(x) =(1/2)(f(x)+f(-x) +f(x)-f(-x))
f(x) =(1/2)(x^2-2x+1+((-x)^2-2(-x)+1) +x^2-2x+1-((-x)^2-2(-x)+1)))....simplify
f(x) =(1/2)(2(x^2-2x+1))
f(x) =x^2-2x+1

Express k(x)=1/(x+1) as the sum of an even and an odd functions

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

 

I am speechless. Great reply! You are a walking book of information....

 

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