A function f(x) : X → Y defined such that

        f(-x) = f(x) ∀ x ε X

is called an even function and

if f (-x) = -d(x) ∀ x ε x, then the function f(x) is called an odd function.

        Graphically, an even function is symmetrical w.r.t. y-axis and odd function is symmetrical w.r.t. origin.

Note :       In general all functions can be represented as sum of an even function and an odd function.

Let, a function be defined as y = f(x). It can be written as:

        => y =  (f(x) + f(-x))/2 + (f(x) - f(-x))/2

        y = F₁(x) + F₂(x)

Whereas,

        F₁(-x) = (f(x) + f(-x))/2 = F₁(x)

And F₂(-x) = (f(-x) - f(x))/2

          = -((f(x) - f(x))/2)

          = -F₂(x).

Here F₁(x) is an even function and F₂(x) is an odd function.

Exercise
State whether the following functions are odd or even or neither.

(1)    y = x³                       

(2)    y = x⁴

(3)    y + x + cos x          

(4)    y = loge(x + √(x² + 1))