Even and Odd Functions: Definition, Examples, Properties

The even and odd functions are classified by their symmetry properties. A function f is even if f(-x)=f(x) and odd if f(-x)=-f(x). In this article, we study even and odd functions with their properties and solve some problems.

Even and odd functions are defined as follows: Even Function: f(-x) = f(x). Odd Function: f(-x) = -f(x). For example, the power function f(x)=xn is an even function if n is even and it is an odd function if n is odd.

Let us now learn even and odd functions in detail.

Definition of Even Function

A function f(x) is said to be an even function if f(-x)=f(x).

For example, f(x)= cosx is an even function as we know cos(-x) = cosx. Also, f(x)=|x| is an example of even function because f(-x) = |-x| = |x| = f(x).

Even Function Examples

Here is a list of some even functions.

• f(x) = x⁰, x², …, x^2m (note x⁰=1 is a constant function)
• Any constant function, for example f(x) = k where k is a constant.
• |x|
• cos x
• sec x
• sin²x.

Definition of Odd Function

A function f(x) is said to be an odd function if f(-x)=-f(x).

For example, f(x)= sinx is an even function as we know sin(-x) = -sinx. Also, f(x) = x³ is an example of an odd function since (-x)³=x³.

Odd Function Examples

The following are few examples of odd functions.

• f(x) = x, x³, …, x^2m+1 (i.e., any odd power of x is an odd function)
• sin x
• cosec x
• tan x
• cot x.

Properties of Even and Odd Functions

The properties of even and odd functions are given in the table below.

Properties	Explanation
1. Even × Even = Even	The product of two even functions is again an even function. For example, the product cosx . secx = 1 is an even function.
2. Odd × Odd = Even	The product of two odd functions is an even function. For example, sinx . cotx = cosx is an even function.
3. Even × Odd = Odd	The product of an even function and an odd function is an odd function. For example, the product cosx . tanx = sinx is an even function.
4. Integral formulas for even and odd functions	$\int_{-a}^a f~dx=$ $2\int_0^a f~dx$ if f is even and the integral is 0 if f is odd.
5. Derivative	The derivative of an even function is odd. The derivative of an odd function is even.
6.	For any function f(x), the sum f(x) + f(-x) is always an even function difference f(x) + f(-x) is an odd function.
7. Sum and difference	The sum and the difference of two even functions (resp. odd functions) are also even (resp. odd).
8. Composition of functions	The composition of two even functions is even. For f(x), g(x) even, the composite function f(g(x)) is even. The composition of an even function and an odd function is even, i.e., f(g(x)) is even provided that f(x) even, g(x) odd or f(x) odd, g(x) even.
9.	Any function f(x) can be expressed as a sum of an even function feven and an odd function fodd, that is, f(x) = feven + fodd where feven = $\dfrac{f(x)+f(-x)}{2}$ and fodd = $\dfrac{f(x)-f(-x)}{2}$.

Mathematical Analysis

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