Odd function

Definition

An odd function is a function that satisfies the property $f(-x) = -f(x)$ for all $x$ in its domain. Graphically, it is symmetric with respect to the origin.

5 Must Know Facts For Your Next Test

1. An odd function integrated over a symmetric interval around zero yields zero: $$\int_{-a}^{a} f(x) \, dx = 0$$.
2. The product of two odd functions is an even function.
3. The sum of two odd functions is also an odd function.
4. If a function is both even and odd, it must be the zero function ($f(x) = 0$).
5. Common examples of odd functions include $f(x) = x^3$, $f(x) = \sin(x)$, and $f(x) = \tan(x)$.

Review Questions

• What is the integral of an odd function over the interval $[-a, a]$?
• If $f(x)$ is an odd function and $g(x)$ is also an odd function, what can you say about their product?
• Given an example of an odd function, verify if it satisfies the condition for being called 'odd'.

Related terms

Even Function: A function such that $f(-x) = f(x)$ for all $x$ in its domain; symmetric with respect to the y-axis.

Symmetry:

A property where one half of something mirrors the other half; in functions, common types include symmetry about the y-axis or origin.

Net Change Theorem:

A theorem stating that the integral of a rate of change (derivative) over an interval gives the net change in the quantity over that interval.