Odd Function-Definition, Properties, and Examples

Odd Function is a type of function that follows the relation f(-x) equals -f(x), where x is any real number in the domain of f(x). This implies that odd functions have the same output for positive and negative input but with an opposite sign. Due to this property, the graph of an odd function is always symmetrical around the origin in cartesian coordinates. Also, this property of odd functions helps one to derive further mathematical relations and get implications for physical quantities expressed by odd functions.

In this article, we will learn about odd functions, their examples, properties, graphical representation of odd functions, some solved examples, and practice questions related to odd functions.

What is an Odd Function?

An Odd Function is a function that has the same magnitude of output for positive and negative input but has opposite signs. In mathematical terms, an odd function follows the relation,

• f(-x) = -f(x)
• f(-x) + f(x) = 0

For all real numbers ‘x’ in the domain of f(x).

Graph of odd functions has a rotational symmetry around the origin in cartesian coordinates. This implies that the graph of odd functions remains unchanged after a rotation of 180^∘ about the origin.

Odd Function Definition

An odd function is defined as a function which follows the relation that f(-x) equals to negative of f(x), for every real number x in the domain of the function.

Odd functions yield the same expression if we substitute -x in place of x in f(x) but with an opposite sign denoted as -f(x). Some examples of odd functions are polynomials involving only odd powers of variables, sin x, tan x, etc.

Odd Function Formula

A function is said to be an odd function if and only if it satisfies the following formula:

f(-x) = -f(x)

For all x ∈ 𝚁 in Domain of Function

Examples of Odd Function

Some examples of odd functions are listed as follows:

• sin x
• tan x
• x²ⁿ⁺¹, where n is a natural number
• sin³x
• tan 3x
• sinh x

Note: To check for an odd function, substitute -x in place of x in the expression of f(x), if the obtained expression is equivalent to -f(x), the function is an odd function otherwise not.

Example: Check whether f(x) = x³ is an odd function or not.

Given,

• f(x) = x³

Substituting -x in place of x in f(x), we get,

⇒ f(-x) = (-x)³ = -x³ = -f(x)

Thus, we get f(-x) = -f(x)

Hence, f(x) is an odd function.

Graph of Odd Function

Graph of an odd function is always symmetrical around the origin in cartesian coordinates, i.e. the plot of the function remains unchanged after a rotation of 180^∘ about the origin. This can be seen in the following image added below:

Properties of an Odd Function

Odd Function have following properties:

• Addition and Subtraction of two odd functions is an odd function.
• Product of two odd functions is an even function.
• Product of an odd function and an even function is an odd function.
• Graph of an odd function exhibits rotational symmetry about origin in cartesian coordinates system.
• Average value of odd functions over a symmetric interval around the origin is zero.
• For any odd function, f(-x) = -f(x).
• Composition of two odd functions is an odd function.
• Definite integral of an odd function about a symmetric interval around origin is zero. i.e.

∫_-a^a f(x).dx = 0

Differences between Odd Function and Even Function

Odd function is a function which follows the properties, f(x) does not equal to f(-x) where as even function are the function which follow the property f(x) equal to f(-x). The basic difference between them is explained in the difference table added below:

Odd Function	Even Function
Odd function is the one in which sign of the output is changed if sign of input is changed but output value remains same.	Even function is the one which doesn’t have any change in output if sign of input is changed.
An odd function follows the relation: f(-x) = -f(x)	An even function follows the relation: f(-x) = f(x)
Graph of an odd function has a rotational symmetry about the origin.	Graph of an even function is symmetrical about Y-axis.
Examples of Odd Functions: x5 , sin x, x, etc.	Examples of Even Functions: x2 , cos x, x6, etc.

Read More,

• Types of Function
• Function
• Trigonometric Functions

Examples on Odd Functions

Example 1: State whether f(x) = x³+2x is an odd function or not?

Solution:

We have, f(x) = x³ +2x

Substituting -x in place of x to get f(-x),

=> f(-x) = (-x)³ + 2(-x) = -x³ – 2x = -(x³ + 2x) = -f(x)

Thus, we get f(-x) = -f(x)

Hence, f(x) is an odd function.

Example 2: Check whether f(x) = x³ + 1 is an odd function or not.

Solution:

We have, f(x) = x³ + 1

To check whether f(x) is an odd function, we need to check for f(-x) = -f(x),

Thus, f(-x) = (-x)³ + 1 = -x³ + 1

And, -f(x) = -(x³ + 1) = -x³ – 1

We see that, f(-x) ≠ -f(x)

Hence, f(x) is not an odd function.

Practice Problems

Problem 1: Check whether f(x) = x³ + x² is an odd function or not.

Problem 2: State whether the function given by f(x) = sin (3x) is an odd function or not?

Problem 3: Prove that, f(x) = 3sin(x) -4sin³x is an odd function.

Problem 4: Determine the value of ‘a’ if f(x) = x³ + a is an odd function.

Problem 5: If f(x) = -|x|, state whether f(x) is an odd function or not.

Odd Function: FAQs

What is meant by an odd function?

A function is said to be an odd function if it follows the relation f(-x) = -f(x), for all real number x in domain of the function.

What are some examples of an odd function?

Some examples of an odd function are x³, sin x, tan x, x⁵, etc.

Which polynomial functions are odd functions?

The polynomials involving only odd powers of variables except any constant in addition or subtraction are odd functions.

Is there any function which is odd as well as even?

Yes, only the zero function, i.e. f(x) = 0 is a function which is even as well as odd because it follows f(-x) = -f(x) = f(x) = 0 for all values of x.

Is sin x an odd function?

Yes, sin x is an odd function as sin(-x) = -sin(x).