Even and Odd functions

A function is even if it is "Symmetric" along the Y-axis both ways.

f(-x) = f(x)

A function is odd if it is "Not Symmetric" along Y-axis but if Negative X part flipped up, it becomes symmetric

f(-x) = -f(x)

When integrated same amount to both sides from $x=0$ , Even functions,

\int_{-a}^{a} f(x)\,dx = 2\cdot \int_{0}^{a} f(x)\,dx

Odd functions,

\int_{-a}^{a} f(x)\,dx = 0

Note

If a function does not hold the definitive properties of Odd or Even, those functions are Neither Even Nor Odd

Pretty Properties

\begin{align*} \text{even} \times \text{even} &= \text {even} \\ \text{odd} \times \text{odd} &= \text {even} \\ \\ \text{odd} \times \text{even} &= \text {odd} \\ \end{align*}

f(x) = odd(x) \pm even(x)

Suppose $f(x)$ is a real valued function. Then,

f(x) = E(x) + O(x) \quad \text{---(1)}

$E(x)$ - Even function, $O(x)$ - Odd function

f(-x) = E(-x) + O(-x)

By definition,

f(-x) = E(x) - O(x)\quad \text{---(2)}

E(x) = \frac{f(x) + f(-x)}{2}

O(x) = \frac{f(x) - f(-x)}{2}

Note

These two function components are unique for $f(x)$ i.e. Only one pair exists