Odd Functions

Odd Functions

We all know Even and Odd numbers.

But what is it about even function? odd function? The output is even? odd?

odd function

Let us read about odd or even function calculator,odd function graph, integral of odd function, odd and even function worksheets, in the concept of Odd functions. Check-out the interactive simulations to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.

Lesson Plan 

Classification of Functions

Functions can be even, odd, both and neither.

Even Function

A function is even if \(f(x) = f( - x)\)  for all \(x\)

This means that the function is the same for \(\text{+ve } x\text{-axis}\) and \(\text{-ve } x\text{-axis,}\) or graphically, symmetric about the \(y\text{-axis}\)

For example, \(f(x) = {x^2}\) 

Even function graph

Odd Function

A function is odd if \( -f(x)=  f(-x)\)  for all \(x\).

The graph of Odd function will be symmetrical about the origin.

For example, \(f(x) = {x^3}\) is odd.

graph of odd function

Neither Even Nor Odd

A function \( f(x)\)  such that\(f(x) \neq  f( - x)\) and  \(-f(x) \neq  f(-x)\)  for all \(x\)   is neither even or odd function.

For example, \(f(x)=2x^{5}+3x^{2}+1\)

Graph of neither odd nor even function

We can see that the graph is not symmetrical about the origin and y-axis.

Both Odd and Even

A function is both odd and even if  \( -f(x)=  f(-x)\)  and  \(f(x) = f( - x)\)  for all \(x\) .

The only function  which is both odd and even is \(f(x) = 0 \)

Both even and odd function graph

The graph is symmetrical about  the origin and y- axis.


Odd Function - Definition

A function is odd if

\(f(-x)= - f(x)\) , for all \(x\).

that is, the function on one side of \(x\text{-axis}\) is sign inverted with respect to the other side or graphically, symmetric about the origin.

Look at the graph of  \(f(x) = x^5\)

Observe the graph in 1st and 3rd quadrant.

The graph looks symmetrical about the origin

odd function graph

Note that all functions  having odd power like are odd functions

\(f(x) = x^7\) is an odd function  but

\(f(x) = x^3 + 2 \) 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

Detrmine the nature of the function \(f(x) = \frac{1}{x}\)

 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 .

\(\begin{align}
f(-x) &= \frac{1}{-x}  \\
&=- \frac{1}{x} \\
& = - f(x)\:\: (\because f(x) = \frac{1}{x})
\end{align}\)

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 \(x\text{-axis}\) 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.

 \(y = x^3 \)

graph of an odd function y = x cube

\(f(x)= -x\) is odd

f(x)=-x graph

\(f(x)=6\sin(x)  \)

graph of 6/sin(x)


Properties of Odd functions

  1. The sum of two odd functions is odd.

  2. The difference of two odd functions is odd.

  3. The product of two odd functions is even.

  4. The quotient of two odd functions is even.

  5. The composition of two odd functions is odd.

  6. The composition of an even function and an odd function is even.

 
tips and tricks
Tips and Tricks
  1. If f(x) is an odd function, then the graph has 180 rotational symmetry about the ORIGIN.
    • Means you could turn it upside-down & it would still look the same.
  2. For every point (x,y) on the graph, the corresponding point (−x,−y) is also on the graph.

Odd or Even Function Calculator

Try your hands on odd and even function calculator to learn about the functions graphically with accuracy.

Solved Examples

Example 1

 

 

Determine algebraically whether the given function \(f(x)=-3 x^{3}+2 x \)s even, odd, or neither.

Solution

\(\text { Let us substitute }-x \text { into the function } f(x)=3 x^{3}+2 x, \text { and then simplify. }\)

A function is odd if \(-f(x) = f(-x) \)

Let us find f(-x)

\begin{aligned}
f(x) &=-3 x^{3}+2 x \\
f (-x) &=-3(-x)^{3}+2(-x) \\
&=-3(-1)^{3}(x)^{3}+2(-x)  \\
&= 3x^3 -2 x  ------- (1) \\
 -f(x) &=- (-3 x^{3}+2 x) \\
&= 3x^3-2x ------ (2)  \\
\end{aligned}
from   (1) and (2)  we  can say that  \(f(-x) = -f(x) \)
and the given function is an odd function.

\(f(x)=-3 x^{3}+2x\) is an odd function.
Example 2

 

 

Determine the nature of the function \( f(x)=x^{3}+2 x\)

Solution

The function is

  • even if \( f(x) = f(−x) \)
  •  odd if \(f(-x) = -f(x) \)

Let us find \(f(-x) \) to determine the nature of the function

 \begin{align}
f(-x)&=(-x)^3+2(-x) \\
&=-x^3-2x \\
 &=  - (x^3 +2x)  \\
 &= -f(x)
\end{align}

\( f(-x)= -f(x)\) this is an odd function
Example 3

 

 

Draw the odd function graph  for the example - 2 \( f(x)=x^{3}+2 x\)   and state  why is it  an odd function.

Solution

Let us plot the given function

odd function graph

  • Notice that the graph is symmetric about the origin .
  • For every point  \((x, y)\) on the graph, the corresponding  point \((-x,-y)\)  is also on the graph
  • For example \((1,3)\) is on the graph of \(f(x)\)  and the corresponding reflection point \((-1,-3)\)   is also on the graph.
Hence the given function \(f(x) is odd\)
Example 4

 

 

Analyse the symmetry of the function \(f(x)=x^{3}-1\)

Solution

\(\)Let us plot the given function

Graph of the given function

The graph of f(x) is neither symmetric to origin nor Y-axis.

We can also solve algebrically,

\begin{align}
f(x)&=x^{3}-1 ----- (1)\\
 f(-x)&=(-x)^{3}-1 \\
&= -x^3 -1\\
f(-x)&= -(x^3 +1) ---- (2) \\
-f(x)&=-(x^{3}-1)\\
&= -x^3 +1 ---- (3)
\end{align}

 we see that   \( f(x) \neq f(−x) \) and \(f(-x) \neq -f(x) \)

The given function is neither even nor odd

Example 5

 

 

Why is the definite integral of an odd function zero?

Solution

The integrand f(x) is an odd function and it is symmetrical about the origin.

We see in most of the odd function graph that the region below and above the x-axis is symmetrical.

We know that, area under the increasing curve is  equal to the area under the decreasing curve.

To prove the definite integral of an odd function zero:

\(\begin{align}
\int_{-a}^{a} f(x) dx
&= \int_{-a}^{0} f(x) dx +\int_{0}^{a} f(x) dx \\
&u = -x \implies du=-dx \\
&\implies u_1= a  ;u_2=0\\
&= -\int_{a}^{0} f(-u) du +\int_{0}^{a} f(x) dx \\
&= -\int_{0}^{a} f(u) du +\int_{0}^{a} f(x) dx \\
&= -\int_{0}^{a} f(x) dx +\int_{0}^{a} f(x) dx \\
&=0
\end{align} \)

 \(\therefore\)definite integral of an odd function zero

Download and practice odd and even function worksheet at the end of this page.

 

Challenge your math skills
Challenging Questions
  1. Determine the nature of the function\(f(-x)=\log(-x+\sqrt{1+(-x)^{2}})\)
  2.  Identify the nature of the function \( f(x+y)=f(x)+f(y) \)for all   \(x, y \in R \)

Interactive Questions

 
 
 
 
 
Download odd and even function worksheets at the end of this page
 

 

 

 

 

 

important notes to remember
Important Notes
  1. A function can be even, odd, neither or both even and odd.
  2. f(x)=0 is the only function that is both even and odd in nature.
  3. A function is odd if \(f(-x) = - f(x)
  4. Graph of Odd function is symmetrical about origin.
  5. A function is odd, if for every point (x,y) on the graph, the corresponding point (−x,−y) is also on the graph.
  6. For an odd function f(x) \(\int_{-a}^{a} f(x) dx = 0 \)

Let's Summarize

The mini-lesson targeted the fascinating concept of odd function. The math journey around odd function starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

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Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


FAQs on Odd Function

How does an odd function look like?

The graph of an odd function is always symmetric about the origin.

Do you know any function which is neither odd or even?

A function can be neither even nor odd. Example:\(x^{3}-x+1\)

How do you tell if a graph is odd, even or neither?

ODD if it is symmetric about origin,
EVEN if it is symmetric about Y-axis
NEITHER if otherwise.

Why is sine an odd function?

We know that \(\sin (-x)=-\sin (x)\).

The graph of sin(x) is symmetric about origin.

So,it is an odd function.

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More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus
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