Even functions and odd functions.

Definition 2.5.8   Let $ f$ be a function defined over a domain $ D$ in $ \mathbb{R}$ . the function $ f$ is even if the two following properties hold:
i.
For every $ x \in D$ , $ -x \in D$ ;
ii.
For every $ x \in D, \; f(-x)=f(x)$ .

Example 2.5.9   Take $ f(x)=x^{2p}$ , where $ p$ is a natural number, i.e. the power is even. The function $ f$ is defined over $ \mathbb{R}$ , therefore the first condition holds. For the second one we have:

$\displaystyle \forall x \in \mathbb{R}, \; f(-x)=(-x)^{2p}=x^{2p}=f(x).$    

Hence, $ f$ is an even function.

Example 2.5.10 (Further elementary even functions)       

Recall that in order to prove that a ``general'' property is not true in a specific situation, a counter-example suffices as a proof.

Example 2.5.11 (Non even functions)       

Definition 2.5.12   Let $ f$ be a function defined over a domain $ D$ in $ \mathbb{R}$ . the function $ f$ is odd if the two following properties hold:
i.
For every $ x \in D$ , $ -x \in D$ ;
ii.
For every $ x \in D, \; f(-x)=-f(x)$ .

Example 2.5.13   Take $ f(x)=x^{2p+1}$ , where $ p$ is a natural number, i.e. the power is odd. The function $ f$ is defined over $ \mathbb{R}$ , therefore the first condition holds. For the second one we have:

$\displaystyle \forall x \in \mathbb{R}, \; f(-x)=(-x)^{2p+1}=x^{2p+1}=f(x).$    

Hence, $ f$ is an odd function.

Example 2.5.14 (Further elementary odd functions)       

Example 2.5.15 (Non odd functions)       

Remark 2.5.16       
  1. There exist functions neither even nor odd, as one of the previous examples shows.
  2. If $ f$ is both even and odd over a domain $ D$ verifying the first condition of Definition 5.8, then $ f$ is the zero function over $ D$ . We leave the proof to the reader.

Remark 2.5.17       
  1. Suppose that $ f$ is an even function over some domain $ D$ . Then the $ y$ -axis is a symmetry axis for the graph of $ f$ (see Figure 8(a)).
  2. If $ g$ is an odd function over some domain $ D$ , then the origin 0 is a symmetry center for the graph of $ f$ (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}
Noah Dana-Picard 2007-12-28