17. Even-odd decompositions
b. If f(x) = f_E(x) + f_O(x) is the sum of an even function f_E(x) and an odd function f_O(x), then show that
f_E(x) = (f(x)+f(-x))/2 and f_O(x) = (f(x)-f(-x))/2

Verified step by step guidance

1. Recall the definitions of even and odd functions: an even function satisfies \(f_E(x) = f_E(-x)\), and an odd function satisfies \(f_O(x) = -f_O(-x)\).
2. Given that \(f(x) = f_E(x) + f_O(x)\), write the expression for \(f(-x)\) by substituting \(-x\) into the function: \(f(-x) = f_E(-x) + f_O(-x)\).
3. Use the properties of even and odd functions to rewrite \(f(-x)\) as \(f(-x) = f_E(x) - f_O(x)\), since \(f_E(-x) = f_E(x)\) and \(f_O(-x) = -f_O(x)\).
4. Add the two equations \(f(x) = f_E(x) + f_O(x)\) and \(f(-x) = f_E(x) - f_O(x)\) to isolate \(f_E(x)\): \(f(x) + f(-x) = 2 f_E(x)\).
5. Similarly, subtract \(f(-x)\) from \(f(x)\) to isolate \(f_O(x)\): \(f(x) - f(-x) = 2 f_O(x)\). Then solve for \(f_E(x)\) and \(f_O(x)\) to get \(f_E(x) = \frac{f(x) + f(-x)}{2}\) and \(f_O(x) = \frac{f(x) - f(-x)}{2}\).

Key Concepts

Even and Odd Functions

An even function satisfies f(x) = f(-x) for all x, meaning its graph is symmetric about the y-axis. An odd function satisfies f(-x) = -f(x), showing symmetry about the origin. Understanding these definitions is essential to decompose any function into even and odd parts.

Function Decomposition

Any function f(x) can be expressed as the sum of an even function f_E(x) and an odd function f_O(x). This decomposition helps analyze the function's symmetry properties and simplifies integration and other operations by separating symmetric components.

Algebraic Manipulation for Decomposition

To find f_E(x) and f_O(x), use the definitions of even and odd functions to form equations involving f(x) and f(-x). Adding and subtracting these equations isolates the even and odd parts, leading to the formulas f_E(x) = (f(x)+f(-x))/2 and f_O(x) = (f(x)-f(-x))/2.