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Odd Function: Definition & Examples

Instructor: Jasmine Cetrone

Jasmine has taught college Mathematics and Meteorology and has a master's degree in applied mathematics and atmospheric sciences.

Odd functions have properties of symmetry that can be useful when graphing and analyzing the functions or when solving equations algebraically. Learn the properties of odd functions and how to recognize them graphically.

What Makes Odd Functions So Special?

Before we race into talking about odd functions, let's have a quick refresher about functions in general! A function describes a relationship between two variables, often x and y, where one of those variables depends on the other. If we say y is a function of x, we are saying that the output value of y depends on the value of x. An important property of a function is that there can be one (and only one) output value. For example, when I ship a package to my friend, the amount that I have to pay to ship the package depends directly on how much that package weighs. When the shipping company weighs the package, they give me exactly one price on how much it is going to cost to ship it. This ensures that shipping cost is a function of package weight.

There are some functions out there with very special relationships between the x- and y-values. For example, let's say that Triangle Man and Circle Girl are were both walking on the Cartesian axes (depicted in the graph below). They both start at the origin, but they always do the opposite of each other. Triangle Man likes to move to the right and Circle Girl to the left. They are also free to move up-and-down as they please, but if Triangle Man goes up 3, Circle girl goes down 3; if he goes down 7, she goes up 7, and so on. After a while their paths create a curve that has origin symmetry, where the positive and negative values of x correspond to opposite values in y. Origin symmetry is representative of a subclass of functions called odd functions.

The Paths of Triangle Man and Circle Girl have Origin Symmetry
Triangle Man and Circle Girl creating a graph with origin symmetry

Another way of looking at the origin symmetry of odd functions is using the idea of reflections. Let's put a curve on the coordinate axis in the first quadrant (graph (a) below), where both x and y are positive. We will then reflect the curve across the y-axis into the second quadrant (graph (b)). Then, we will take that blue reflected curve and use the x-axis as a mirror, reflecting the blue curve into the third quadrant to make the green curve (graph (c)). If we erase the blue curve that we had in quadrant two, and leave the rest (graph (d)), we have a curve that now has origin symmetry--the left-hand side is a double reflection (horizontally and vertically) of the right-hand side.

The final graph displays odd symmetry
creating an odd function with symmetry

Defining Odd Functions

There are two ways to describe odd functions - graphically and algebraically.

If we look at the graph of an odd function, we will notice that it has origin symmetry. The x and y axes together form a set of mirrors to make the graph look the same across the origin. Graphs of functions with origin symmetry are shown below.

Graphs of functions that display symmetry about the origin are odd functions
graphs that have origin symmetry are even

These graphs, as well as the graph formed by our friends Triangle Man and Circle Girl, can help us build the algebraic definition of an odd function: the y-value of the function at negative x is always opposite sign as the y-value of the function at positive x. Or, algebraically, an odd function is a function such that

f(-x)=-f(x)

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