EVEN AND ODD FUNCTION

 

Even functions:

f(- x) = f(x)
 

Even function is symmetric with respect to the y-axis.

It means that the graph remains unchanged after reflection about the y-axis.


Examples of even functions:

f(x) = 2
 

f(x) = |x|

 

 

f(x) = x2 f(x) = x4

 

 

f(x)=cos x f(x)= e^(-x^2)

 

 

f(x) = x2 + |x| f(x) = x2 - 3

 

 

Odd functions:

f(- x) = − f(x)

Odd function is symmetric with respect to the origin.

It means that its graph remains unchanged after rotation of 180 degrees about the origin.

Examples of odd functions:
 

f(x) = x f(x) = x3

 

 

f(x) = sin x f(x) = tan x

 

 

f(x) =        f(x) = 4x
 

Neither:

If  f(- x) ≠ f(x)  and  f(- x) ≠ − f(x), the function is neither even nor odd.

Below are given some examples:

f(x) = x2-8x+9

f(x) = x2+x

 

 

  

 

Properties of Even and Odd Functions:

The sum of an even and odd function is neither even nor odd.  E.g:  f(x)= x2 + x

The sum of two even functions is even.  E.g:  f(x)= 2x2 + x2 = 3x2

The sum of two odd functions is odd.   E.g:  f(x)= x3 + x

Any constant multiple of an even function is even.  E.g:  f(x)= x2,     g(x)= 5x2

Any constant multiple of an odd function is odd.  E.g 2: f(x)= x3,    g(x)= 4x3    

The product of two even functions is an even function. E.g:  f(x)= x4 * x2 = x6

The product of two odd functions is an even function.
E.g:  f(x)= x3 * x = x4

The product of an even function and an odd function is an odd function.  E.g:  f(x)= x2 * x = x3

The quotient of two even functions is an even function.  E.g:  f(x)= x6 : x2 = x4

The quotient of two odd functions is an even function.  f(x)= x3 : x = x2

The quotient of an even function and an odd function is an odd function.  f(x)= x4 * x = x3

The derivative of an even function is odd.  f(x) = x4 ,           f'(x)= 4x3

The derivative of an odd function is even.  f(x) = x5 ,           f'(x)= 5x4