Odd Functions
We all know Even and Odd numbers.
But what is it about even function? odd function? The output is even? odd?
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. Checkout 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, \(f(x) = {x^2}\)
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.
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, \(f(x)=2x^{5}+3x^{2}+1\)
We can see that the graph is not symmetrical about the origin and yaxis.
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 \(f(x) = 0 \)
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 1^{st }and 3^{rd} quadrant.
The graph looks symmetrical about the origin
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.
\(y = x^3 \)
\(f(x)= x\) is odd
\(f(x)=6\sin(x) \)
Properties of Odd functions

The sum of two odd functions is odd.

The difference of two odd functions is odd.

The product of two odd functions is even.

The quotient 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.
 If f(x) is an odd function, then the graph has 180 rotational symmetry about the ORIGIN.
 Means you could turn it upsidedown & it would still look the same.
 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^32x  (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^32x \\
&=  (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
 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
The graph of f(x) is neither symmetric to origin nor Yaxis.
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 xaxis 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.
 Determine the nature of the function\(f(x)=\log(x+\sqrt{1+(x)^{2}})\)
 Identify the nature of the function \( f(x+y)=f(x)+f(y) \)for all \(x, y \in R \)
Interactive Questions
 A function can be even, odd, neither or both even and odd.
 f(x)=0 is the only function that is both even and odd in nature.
 A function is odd if \(f(x) =  f(x)
 Graph of Odd function is symmetrical about origin.
 A function is odd, if for every point (x,y) on the graph, the corresponding point (−x,−y) is also on the graph.
 For an odd function f(x) \(\int_{a}^{a} f(x) dx = 0 \)
Let's Summarize
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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 Yaxis
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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