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 counter-example 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.
![\begin{figure}\centering
\mbox{
\subfigure[even function]{\epsfig{file=EvenFunct...
...bfigure[odd function]{\epsfig{file=OddFunction.eps,height=5cm}}
}\end{figure}](img303.png) |
Noah Dana-Picard
2007-12-28