Odd Function - Definition, Properties, Graph, Examples - Cuemath

Odd Function

The odd functions are functions that return their negative inverse when x is replaced with –x. This means that f(x) is an odd function when f(-x) = -f(x). Some examples of odd functions are trigonometric sine function, tangent function, cosecant function, etc. Let us understand the odd functions and their properties in detail in the following section.

1.	What is an Odd Function?
2.	General formula of an ODD Function
3.	Graphical Representation of Odd Function
4.	Properties of Odd Functions
5.	FAQs on Odd Functions

What is an Odd Function?

A function is odd if −f(x) = f(−x), for all x. The graph of an odd function will be symmetrical about the origin. For example, f(x) = x3 is odd. That is, the function on one side of x-axis is sign inverted with respect to the other side or graphically, symmetric about the origin.

Odd Function Example

Look at an example of an odd function, a graph of f(x) = x5

Observe the graph in the 1st and 3rd quadrants. The graph looks symmetrical about the origin. Note that all functions having odd power like are odd functions.

f(x) = x7 is an odd function but f(x) = x3 + 2 is not an odd function. Can you verify with the odd function rule?

General formula of an Odd Function

Algebraically, without looking at a graph, we can determine whether the function is even or odd by finding the formula for the reflections.

f(−x) = −f(x) for all x

Example:

Determine the nature of the function f(x) = 1/x

The function is odd, if f(−x) = −f(x) and even if f(x) = f(−x),

Let us find f(−x) to determine the nature of the function.

f(−x) = 1/-x = −1/x = −f(x) (∵ f(x) = 1/x)

Since f(−x) = −f(x) the function is odd.

Graphical Representation of Odd Function

Odd Functions are symmetrical about the origin. The function on one side of x-axis is sign inverted with respect to the other side or graphically, symmetric about the origin. Here are a few examples of odd functions, observe the symmetry about the origin.

y = x3

f(x) = −x is odd

f(x) = 6sin(x)

Properties of Odd Functions

Like other functions in maths, odd functions have their own properties which can b used to identify the odd function easily. Let us look at few properties.

• The sum of two odd functions is odd.
• The difference between two odd functions is odd.
• The product of two odd functions is even.
• The quotient of the division of two odd functions is even.
• The composition of two odd functions is odd.
• The composition of an even function and an odd function is even.

☛Articles on Odd Function

Given below is the list of topics that are closely connected to the odd function. These topics will also give you a glimpse of how such concepts are covered in Cuemath.

• Exponential Function
• Polynomial Functions
• Quadratic Functions
• Linear Functions
• Constant Functions