Odd Functions

Definition

A function $f$ is odd if $f(-x) = -f(x)$ for every $x$ in its domain. The graph of an odd function is symmetric about the origin.

• How to read: “The function evaluated at negative x equals negative f of x.”
• Meaning: Negating the input flips the sign of the output—180° rotational symmetry about the origin.

Why It Matters

Identifying odd symmetry is a powerful tool in calculus; for example, the integral of an odd function over a symmetric interval $[ -a, a ]$ is always zero. This property simplifies complex calculations in physics and engineering, particularly in wave mechanics and Fourier analysis.

Core Concepts

• Symmetry Test: To test if a function is odd, substitute $-x$ for $x$ and simplify. If the result is the negation of the original function, it is odd.
• Polynomials: Polynomials with only odd powers of $x$ are odd functions.
  • Example: $f(x) = x$ is an odd function because $(-x) = -x$.
    • How to read: “The function evaluated at negative x equals negative x, where f of x equals x.”
    • Meaning: Identity with sign flip—simplest odd function.
• Trigonometric Functions: The sine function is a primary example of an odd function.
  • Example: $f(x) = \sin x$ is an odd function.
    • How to read: “The sine of negative x equals negative sine of x.”
    • Meaning: Sine flips sign under x → −x—origin symmetry.

Connected Concepts

• Function Definition, Function Notation
• Function Domain
• Function Range
• Graphs of Functions: Properties
• Even Functions