What is the defining property of an odd function?

An odd function is characterized by a specific symmetry known as rotational symmetry about the origin. This means that if a point (x, y) lies on the graph of the function, then the point (-x, -y) also lies on the graph. Mathematically, this property can be expressed as f(-x) = -f(x) for all x in the function's domain. This symmetry about the origin leads to the characteristic shape of odd functions, which often appear to be mirrored across both axes.

For instance, if you take the classic example of the cubic function f(x) = x³, you'll notice that if you evaluate f(1) = 1 and f(-1) = -1, the output values are opposites, confirming that the function is odd. The defining property of being symmetric about the origin is central to understanding the behavior of odd functions across different quadrants of the Cartesian plane.

Other options such as symmetry about the y-axis describe even functions instead, constant values do not pertain to any specific type of function, and functions with only positive outputs do not define odd functions since odd functions can output both positive and negative values. Thus, understanding the symmetry about the origin is critical in identifying and working with