## 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.

Example 2.5.10 (Further elementary even functions)
• A constant function over is even;
• The absolute value function is even over ;
• The cosine function is even over ;
• If , where is a negative integer, then is an even function over .

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)
• Take . The function is defined over , thus the first condition of Definition 5.8 holds. But and , thus ; this means that is not even.
• Let . The function is defined over . The first condition in Definition 5.8 is not verified, as is in the domain, but is not. Therefore is not even.

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.

Example 2.5.14 (Further elementary odd functions)
• The sine function is odd;
• The tangent function is odd (over ;
• If , where is a negative integer, then is an odd function over .

Example 2.5.15 (Non odd functions)
• Take . The function is defined over , thus the first condition of Definition 5.8 holds. But and , thus ; this means that is not odd.
• Let . The function is defined over . The first condition in Definition 5.8 is not verified, as is in the domain, but is not. Therefore is not odd.

Remark 2.5.16
1. There exist functions neither even nor odd, as one of the previous examples shows.
2. 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
1. Suppose that is an even function over some domain . Then the -axis is a symmetry axis for the graph of (see Figure 8(a)).
2. If is an odd function over some domain , then the origin 0 is a symmetry center for the graph of (see Figure 8(b)).

Noah Dana-Picard 2007-12-28