Even and odd functions

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an even function is a <a href="/facts/Real_function/bbNG3DnN">real function</a> such that 
 
 
 
 f
 (
 −
 x
 )
 =
 f
 (
 x
 )
 
 
 {\displaystyle f(-x)=f(x)}
 
 for every 
 
 
 
 x
 
 
 {\displaystyle x}
 
 in its <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a>. Similarly, an odd function is a function such that 
 
 
 
 f
 (
 −
 x
 )
 =
 −
 f
 (
 x
 )
 
 
 {\displaystyle f(-x)=-f(x)}
 
 for every 
 
 
 
 x
 
 
 {\displaystyle x}
 
 in its domain.
They are named for the <a href="/facts/Parity_(mathematics)/7zq2UM4V">parity</a> of the powers of the <a href="/facts/Power_Function/pac6QY2g">power functions</a> which satisfy each condition: the function 
 
 
 
 f
 (
 x
 )
 =
 
 x
 
 n
 
 
 
 
 {\displaystyle f(x)=x^{n}}
 
 is even if n is an <a href="/facts/Even_integer/7zq2UM4V">even integer</a>, and it is odd if n is an odd integer.
Even functions are those real functions whose <a href="/facts/Graph_of_a_function/htYX0BGX">graph</a> is <a href="/facts/Symmetry_(geometry)/TljpwtNQ">self-symmetric</a> with respect to the y-axis, and odd functions are those whose graph is self-symmetric with respect to the <a href="/facts/Origin_(mathematics)/Lm26tRCB">origin</a>.
If the domain of a real function is self-symmetric with respect to the origin, then the function can be uniquely decomposed as the sum of an even function and an odd function.

Even and odd functions open-in-new

Even and odd functions