y = x^{2/3}

Graph of the Power Function $y = x^{2/3}$

$y = x^{2/3}$ is a power function with a fractional exponent, and can be written $x^{2/3} = \left(\sqrt[3]{x}\right)^2 = \sqrt[3]{x^2}$ . Combining a cube root with a square, and because $x^2$ is always non-negative, it is defined as a real number even for negative $x$ .

Domain and symmetry

The domain is all real numbers
The range is $y \geq 0$
Even function
The minimum $0$ is attained at the origin

Since $(-x)^{2/3} = \sqrt[3]{x^2} = x^{2/3}$ , the graph is symmetric about the $y$ -axis.

The cusp at the origin

The derivative is $y' = \dfrac{2}{3\sqrt[3]{x}}$ .

Approach	$y'$
$x \to 0^{+}$	$\to +\infty$
$x \to 0^{-}$	$\to -\infty$

The tangents on both sides become vertical at the origin. Such a sharp point is called a cusp¹. In contrast to the parabola $y = x^2$ , which is smooth at the origin, $x^{2/3}$ plunges into the origin even more sharply than a V. Because the one-sided slopes are infinite and disagree, the function is not differentiable there.

Compared with the cube root

Function	Parity	The origin
$\sqrt[3]{x}$	odd	inflection with a vertical tangent
$x^{2/3}$	even	cusp, and the minimum

Squaring the cube root folds everything up onto the non-negative side, which is what turns the inflection into a cusp.

Notable points

$x$	$x^{2/3}$
$\pm 1$	$1$
$\pm 8$	$4$
$\pm 27$	$9$

As $x \to \pm\infty$ the values also grow without bound. As a power function $x^p$ with $0 < p < 1$ , it rises steeply near the origin and gently far away.

Where the shape appears

This shape appears in the famous astroid.

x^{2/3} + y^{2/3} = a^{2/3}

Two-thirds-power relationships also occur in natural scaling laws. Kepler's third law says the square of the orbital period is proportional to the cube of the semi-major axis², so recovering the axis from the period is exactly a $\dfrac{2}{3}$ power.

Singular point of a curve, Wikipedia
Kepler's laws of planetary motion, Wikipedia