EVEN AND ODD FUNCTION
Even functions:
f( x) = f(x)
Even function is symmetric with respect to the yaxis.
It means that the graph remains unchanged after reflection about the yaxis.
Examples of even functions:
f(x) = 2 
f(x) = x 


f(x) = x^{2}  f(x) = x^{4} 


f(x)=cos x  f(x)= 


f(x) = x^{2 }+ x  f(x) = x^{2 } 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) = x^{3} 


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) = x^{2}8x+9 
f(x) = x^{2}+x 


Properties of Even and Odd Functions:
The sum of an even and odd function is neither even nor odd. E.g: f(x)= x^{2} + x
The sum of two even functions is even. E.g: f(x)= 2x^{2} + x^{2 = }3x^{2}
The sum of two odd functions is odd. E.g: f(x)= x^{3 + } x
Any constant multiple of an even function is even. E.g: f(x)= x^{2}, g(x)= 5x^{2}
Any constant multiple of an odd function is odd. E.g 2: f(x)= x^{3}, g(x)= 4x^{3}
The product of two even functions is an even function. E.g: f(x)=
x^{4 }*^{ }
x^{2}
= x^{6 }
The product of two odd functions is an even function. E.g: f(x)=
x^{3 * }
x =
x^{4 }
The product of an even function and an odd function is an odd function. E.g: f(x)= x^{2 * } x = x^{3}
The quotient of two even functions is an even function. E.g: f(x)= x^{6 }:^{ } x^{2} = x^{4}
The quotient of two odd functions is an even function. f(x)= x^{3 : } x = x^{2 }
The quotient of an even function and an odd function is an odd function. f(x)= x^{4 * } x = x^{3 }
The derivative of an even function is odd. f(x) = x^{4} , f'(x)= 4x^{3}
The derivative of an odd function is even. f(x) = x^{5} , f'(x)= 5x^{4}