Chapter 1.8, Problem 73E

Proof

(a) Given a function f, prove that g is even and h is odd, where $g\left(x\right)=\frac{1}{2}\left[f\left(x\right)+f\left(-x\right)\right]\text{ and }h\left(x\right)=\frac{1}{2}\left[f\left(x\right)-f\left(-x\right)\right].$

(b) Use the result of part (a) to prove that any function can be written as a sum of even and odd functions. [Hint: Add the two equations in part (a).]

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

$f\left(x\right)=x^2-2x+1,k\left(x\right)=\frac{1}{x+1}$