Odd function

Definition

An odd function is a function $f(x)$ that satisfies the condition $f(-x) = -f(x)$ for all $x$ in its domain. The graph of an odd function is symmetric about the origin.

5 Must Know Facts For Your Next Test

1. If $f(x)$ is an odd function, then the integral of $f(x)$ from $-a$ to $a$ is zero, i.e., $\int_{-a}^{a} f(x) \, dx = 0$.
2. The sum of two odd functions is also an odd function.
3. The product of two odd functions is an even function.
4. If a function has rotational symmetry around the origin (180-degree rotation), it is an odd function.
5. A polynomial function with only odd powers of $x$ (like $x^3$, $x^5$, etc.) and no constant term is always an odd function.

Review Questions

• What condition must a function satisfy to be considered an odd function?
• How does the integral of an odd function over a symmetric interval behave?
• Give an example of a polynomial that represents an odd function.

Related terms

Even Function: $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in its domain, and its graph is symmetric about the y-axis.

Symmetry:

A property where one part of a figure or graph can be transformed into another part by some operation such as reflection or rotation.

$\int_{-a}^{a} f(x) \, dx$: This represents the definite integral of a function from $-a$ to $a$. For an odd function, this value equals zero.