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 ${\displaystyle x^{2}}$ , ${\displaystyle x^{4}}$ , ${\displaystyle cos(x)}$ , ${\displaystyle cosh(x)}$ .

• The double derivative of any even function is an even function.
• The product of 2 even functions is an even function.
• The sum of 2 even functions is an even function.

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 ${\displaystyle x}$ , ${\displaystyle x^{3}}$ , ${\displaystyle sin(x)}$ , ${\displaystyle sinh(x)}$ .

See also

• Taylor_series