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:

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

Hence, $f$ is an even function.

Example 2.5.10 (Further elementary even functions)       

• A constant function over $\mathbb{R}$ is even;
• The absolute value function is even over $\mathbb{R}$ ;
• The cosine function is even over $\mathbb{R}$ ;
• If $f(x)=x^{2p}$ , where $p$ is a negative integer, then $f$ is an even function over $\mathbb{R}-\{ 0 \}$ .

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 $f(x)=2x+1$ . The function $f$ is defined over $\mathbb{R}$ , thus the first condition of Definition 5.8 holds. But $f(1)=3$ and $f(-1)=-1$ , thus $f(1)\neq f(-1)$ ; this means that $f$ is not even.
• Let $g(x)=1/(x-1)$ . The function $g$ is defined over $\mathbb{R}-\{ 1 \}$ . The first condition in Definition 5.8 is not verified, as $-1$ is in the domain, but $1$ is not. Therefore $f$ is not even.

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:

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

Hence, $f$ is an odd function.

Example 2.5.14 (Further elementary odd functions)       

• The sine function is odd;
• The tangent function is odd (over $\mathbb{R}- \{ \pi /2 + k \pi; \; k \in \mathbb{Z} \}$ ;
• If $f(x)=x^{2p+1}$ , where $p$ is a negative integer, then $f$ is an odd function over $\mathbb{R}-\{ 0 \}$ .

Example 2.5.15 (Non odd functions)       

• Take $f(x)=2x+1$ . The function $f$ is defined over $\mathbb{R}$ , thus the first condition of Definition 5.8 holds. But $f(1)=3$ and $f(-1)=-1$ , thus $f(-1)\neq -f(1)$ ; this means that $f$ is not odd.
• Let $g(x)=1/(x-1)$ . The function $g$ is defined over $\mathbb{R}-\{ 1 \}$ . The first condition in Definition 5.8 is not verified, as $-1$ is in the domain, but $1$ is not. Therefore $f$ is not odd.

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.