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. This symmetry about the origin means that rotating the graph 180 degrees around the origin will produce the same graph.

5 Must Know Facts For Your Next Test

1. The graph of an odd function is symmetric with respect to the origin.
2. If a function is odd, then $f(0) = 0$ must hold true if $0$ is in its domain.
3. Common examples of odd functions include $y = x^3$, $y = \sin(x)$, and $y = \tan(x)$.
4. Verifying whether a trigonometric identity results in an odd function can help simplify solving trigonometric equations.
5. In transformations, applying a horizontal or vertical flip to an odd function still results in an odd function.

Review Questions

• What condition must be satisfied for a function to be considered an odd function?
• Give two examples of common trigonometric functions that are odd functions.
• How does symmetry play a role in identifying an odd function?

Related terms

Even Function: A function $f(x)$ is even if it satisfies $f(-x) = f(x)$ for all $x$ in its domain, indicating symmetry with respect to the y-axis.

Symmetry: In mathematics, symmetry refers to a property where one part of a figure or object mirrors another part. In functions, common symmetries include y-axis (even functions) and origin (odd functions).

\(\sin(x)\): \(\sin(x)\) is an example of an odd trigonometric function because \(\sin(-x) = -\sin(x)\).