Even functions and odd functions.
Definition 2.5.8
Let
be a function defined over a domain
in
. the function
is
even if the two following properties hold:
 i.
 For every
,
;
 ii.
 For every
.
Example 2.5.9
Take
, where
is a natural number, i.e. the power is even. The function
is defined over
, therefore the first condition holds. For the second one we have:
Hence,
is an even function.
Recall that in order to prove that a ``general'' property is not true in a specific situation, a counterexample suffices as a proof.
Definition 2.5.12
Let
be a function defined over a domain
in
. the function
is
odd if the two following properties hold:
 i.
 For every
,
;
 ii.
 For every
.
Example 2.5.13
Take
, where
is a natural number, i.e. the power is odd. The function
is defined over
, therefore the first condition holds. For the second one we have:
Hence,
is an odd function.
Remark 2.5.16
 There exist functions neither even nor odd, as one of the previous examples shows.
 If
is both even and odd over a domain
verifying the first condition of Definition 5.8, then
is the zero function over
. We leave the proof to the reader.
Remark 2.5.17
 Suppose that
is an even function over some domain
. Then the
axis is a symmetry axis for the graph of
(see Figure 8(a)).
 If
is an odd function over some domain
, then the origin 0 is a symmetry center for the graph of
(see Figure 8(b)).
Figure 8:
The graph of an either even or odd function.

Noah DanaPicard
20071228