Even and odd functions

Even functions

An even function is a function which satisfies the condition:

${\displaystyle f(-x)=f(x)}$ 

Therefore an even function is symmetric with respect to the y-axis.

The denomination even is due to the fact that the Taylor series of an even function includes only even powers.

Examples of even functions are:

• x⁽²ⁿ⁾ where n is any non-negative integer
• ${\displaystyle \cos(x)}$ 
• ${\displaystyle \cosh(x)}$ 
• The absolute value
• The double derivative of any even function
• The composition of any 2 functions, at least one of which is even
• The product of 2 even functions
• The sum of 2 even functions

Odd functions

An odd function is a function which satisfies the condition:

${\displaystyle f(-x)=-f(x)}$ 

Therefore an odd function is symmetric with respect to the origin.

The denomination odd is due to the fact that the Taylor series of an odd function includes only odd powers.

Examples of odd functions are ${\displaystyle x^{3}}$  and ${\displaystyle \sin x}$ 

See also

• Taylor_series