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)=
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 + x^{2 =} 3x²

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)= 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)= x^{3 *} x = x⁴

The product of an even function and an odd function is an odd function.  E.g:  f(x)= x^{2 *} 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)= x^{3 :} x = x²

The quotient of an even function and an odd function is an odd function.  f(x)= x^{4 *} 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