Odd Functions : Definition , Graph ,examples

What is odd functions

An odd function is a function f(x) that satisfies the property f(-x) = -f(x) for all x in the domain of the function.
In other words, an odd function is symmetric with respect to the origin.
Geometrically, an odd function has the property that if you rotate the graph of the function by 180 degrees about the origin, the graph looks the same
Graph of odd function is diatonically symmetrical in opposite quadrant
If the function is not odd, then it can be even or neither odd nor even

Examples

(1) Polynomial functions

Any polynomial with only odd degree terms is an odd function, for example, f(x) = 2x⁷ + 9x⁵ – x. (Note that all the powers of x are odd numbers.)

(2)Sine function: sin(x)

The sine function is defined as the ratio of the opposite side to the hypotenuse in a right-angled triangle. It is an odd function because sin(-x) = -sin(x) for all x. This property can also be derived from the geometric interpretation of sine, which is that it represents the y-coordinate of a point on the unit circle.

(3) Cubic function: $x^3$

The cubic function is defined as $f(x) = x^3$. It is an odd function because f(-x) = -f(x) for all x. This property can be seen by observing that the graph of the function is symmetric about the origin.

(4) Sign function: sign(x)

The sign function is defined as f(x) = x/|x| for all nonzero values of x, and f(0) = 0. It is an odd function because f(-x) = -f(x) for all x. This can be seen by considering the cases where x is positive, negative, or zero.

Properties of odd functions

(1) For any function f(x), h(x) =f(x) + f(-x) is an odd function.

h(-x) = f(-x) + f(x) = -f(x) -f(-x)= -[f(x) + f(-x)]=-h(x)

(2) The sum or difference of two odd functions is odd

let f(x)= -f(-x) and g(x) = -g(-x)

Now h(x) = f(x) + g(x)
h(-x) = f(-x) +g(-x) =-f(x) – g(x)= -h(x)

(3) The multiple of an odd function is again an odd function.

let f(x)= -f(-x) and h(x) = n f(x)
then
h(-x) = nf(-x) = -nf(x) = -h(x)

(4) The product or division of two odd functions is even.

let f(x)=- f(-x) and g(x) = -g(-x)
Now h(x) = f(x) g(x)
h(-x) = f(-x) g(-x) =f(x) g(x)= h(x)

(5) The composition of two odd functions is odd

let Both even f(x)= -f(-x) and g(x) =- g(-x)

$f \circ g (x) = f(g(x)$
Now $f \circ g (-x)= f(g(-x)=- f(g(x)$

How to Prove the odd Function

We simply need to prove that f(x) =- f(-x)

Solved Examples

Example 1

$f(x) = x^5 -x^3 -2x$

Solution

$f(-x) =(-x)^5 -(-x)^3 -2(-x)=-x^5 +x^3 +2x= -(x^5 -x^3 -2x) =-f(x)$

Hence an odd function

Example 2

$f(x)= \frac { \sqrt {1+x} + \sqrt {1-x}}{\sqrt {2+x} – \sqrt {2-x}}$

Solution

$f(-x) =\frac { \sqrt {1-x} + \sqrt {1+x}}{\sqrt {2-x} – \sqrt {2+x}}= -\frac { \sqrt {1+x} + \sqrt {1-x}}{\sqrt {2+x} – \sqrt {2-x}}= -f(x) $

Hence odd function