is the third-degree monomial obtained by multiplying by itself three times1. It flattens out for an instant at the origin, then climbs without bound to the upper right.
Because the exponent is odd, a negative gives a negative : unlike , this function takes negative values.
From , it is an odd function with point symmetry about the origin. Only the sign flips, so gives while gives . The graph lies in the first and third quadrants.
The derivative is , positive everywhere except at , so the function increases monotonically over the whole line. At the derivative vanishes and the tangent line lies along the -axis, but the function keeps increasing on both sides, so this is neither a maximum nor a minimum.
The second derivative changes sign at , which makes the origin an inflection point: the curve is concave down for and concave up for .
The curve flattens near the origin and shoots up once exceeds . Where is even and never negative, is odd and preserves sign.
Since is a one-to-one map from the reals onto the reals, it has an inverse: the cube root . The equation has exactly one real solution for every real .
It is the archetype of the odd powers , and beyond. Every odd power passes through , and .
Doubling the side of a cube multiplies its volume by : the volumes of similar solids grow with the cube of length.