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