odd function

An odd function has a spinning kind of symmetry: rotate its graph half a turn about the origin and it lands exactly on itself. Where the curve rises on the right, it sinks by the same amount on the left, mirrored through the center point.

The algebraic test is f(-x) = -f(x) for every x in the domain. Flipping the sign of the input flips the sign of the output too. The name again comes from powers: x, x^3, x^5 and other odd powers are odd functions, because raising -x to an odd power produces the negative of x raised to that power.

A useful consequence: if an odd function is defined at 0, then f(0) must equal 0, because f(0) = -f(0) forces the value to be zero. As with even functions, most functions are neither even nor odd, so do not assume — test by substituting -x and comparing.