Graph of the Cube Root Function $y = \sqrt[3]{x}$

$y = \sqrt[3]{x}$ is the number whose cube is $x$; it can also be written $x^{1/3}$¹. Cubing keeps the sign of a negative number, so negative inputs have real cube roots too: $\sqrt[3]{-8} = -2$.

Domain and range

• Both the domain and the range are all real numbers
• The function increases monotonically
• Odd function
• The origin is an inflection point with a vertical tangent

This mirrors the fact that $x^3$ is a one-to-one map from the reals onto the reals: its inverse returns exactly one real value for every real input. The square root, by contrast, is defined only for $x \geq 0$.

Symmetry

Since $\sqrt[3]{-x} = -\sqrt[3]{x}$, it is an odd function with point symmetry about the origin, and its graph spreads through the first and third quadrants.

Monotonicity and the origin

The derivative is $y' = \dfrac{1}{3\sqrt[3]{x^2}}$, positive for every $x \neq 0$, so the function increases over the whole line. As $x \to 0$ the derivative tends to $+\infty$, so the tangent at the origin is vertical and the function is not differentiable there.

The origin is an inflection point, and the graph takes an S-like shape around its vertical tangent.

Notable points

It takes an eightfold increase in $x$ to double $y$, and a thousandfold increase to multiply $y$ by ten: steep near the origin, extremely gentle far away.

Relationships with other functions

It is the inverse of $y = x^3$, and the two graphs are reflections of each other in the line $y = x$. Squaring it gives $x^{2/3}$, an even function with a sharp cusp at the origin, a markedly different shape.

Applications

• A cube of volume $V$ has side $\sqrt[3]{V}$
• Recovering a length from a volume among similar solids
• Biological scaling laws that estimate body length from body mass
• Cardano's formula for solving cubic equations

1. Cube root, Wikipedia