An example of an odd function is $f(x) = x^3$.

Equation:

The function is odd because it satisfies the property of odd functions, which is: $f(-x) = -f(x)$ Let's check this for $f(x) = x^3$:

• $f(x) = x^3$
• $f(-x) = (-x)^3 = -x^3 = -f(x)$

Thus, $f(x) = x^3$ is an odd function because $f(-x) = -f(x)$.

From its Graph:

An odd function exhibits symmetry about the origin. This means that for any point $(x, y)$ on the graph, the point $(-x, -y)$ must also be on the graph. For $f(x) = x^3$, the graph has this symmetry. For instance:

• At $x = 1$, $f(1) = 1^3 = 1$.
• At $x = -1$, $f(-1) = (-1)^3 = -1$.

You can observe that the graph reflects across the origin.

From its Table:

If you look at a table of values, you'll notice that for every $x$, the function at $-x$ is the negative of the function at $x$. For example:

$x$	$f(x) = x^3$
-2	-8
-1	-1
0	0
1	1
2	8

Notice that $f(-2) = -f(2)$, $f(-1) = -f(1)$, and so on, which confirms the odd nature of the function.

Summary:

• Equation: $f(-x) = -f(x)$ holds true.
• Graph: The function is symmetric about the origin.
• Table: The function values at $x$ and $-x$ are opposites.

Questions:

1. Can you give an example of an even function and how it differs from an odd one?
2. What happens when you add an odd and an even function?
3. How would the graph of $f(x) = -x^3$ differ from $f(x) = x^3$?
4. Are all polynomial functions either even or odd?
5. What happens if you multiply two odd functions?

Tip:

To check whether a function is odd, always test if $f(-x) = -f(x)$. If it holds for all $x$, the function is odd.