y = \dfrac{1}{x^3}

Graph of the Function $y = 1/x^3$

$y = \dfrac{1}{x^3}$ is the power function $y = x^n$ with $n = -3$ . Like the reciprocal it is symmetric about the origin, but the higher power in the denominator makes it rise more steeply near the origin and fall toward $0$ more quickly far away.

Domain and range

The domain is $x \neq 0$
The range is $y \neq 0$
Odd function
Each branch decreases, with no extrema

Symmetry

From $f(-x) = -f(x)$ the function is odd. It is positive for $x > 0$ and negative for $x < 0$ , so the curve lies only in the first and third quadrants. This contrasts with the even power $y = \dfrac{1}{x^2}$ , which occupies the first and second quadrants.

Function	Parity	Quadrants
$\dfrac{1}{x^2}$	even	first and second
$\dfrac{1}{x^3}$	odd	first and third

Monotonicity

The derivative is $f'(x) = -\dfrac{3}{x^4}$ . Since $x^4$ is positive whenever $x \neq 0$ , the derivative is always negative, and the function decreases on each branch separately. It is not decreasing across the whole domain, however: $f(-1) = -1$ while $f(1) = 1$ , so the value actually rises as you step over $x = 0$ .

Concavity

The second derivative is $f''(x) = \dfrac{12}{x^5}$ , positive for $x > 0$ and negative for $x < 0$ . The right half is concave up and the left half concave down. The sign changes at $x = 0$ , but that point is outside the domain, so there is no inflection point.

Asymptotes

As $x \to 0^{+}$ we have $y \to +\infty$ and as $x \to 0^{-}$ we have $y \to -\infty$ , so the $y$ -axis is a vertical asymptote. As $x \to \pm\infty$ we have $y \to 0$ , making the $x$ -axis a horizontal asymptote.

Comparison with the reciprocal

The two graphs meet at $(1, 1)$ and $(-1, -1)$ , since solving $\dfrac{1}{x^3} = \dfrac{1}{x}$ gives $x^2 = 1$ .

$x$	$\dfrac{1}{x}$	$\dfrac{1}{x^3}$
$\dfrac{1}{2}$	$2$	$8$
$1$	$1$	$1$
$2$	$0.5$	$0.125$

For $|x| > 1$ the cube is closer to zero, while for $0 < |x| < 1$ it swings much further.

Inverse function

Solving $y = \dfrac{1}{x^3}$ for $x$ gives the inverse $y = \dfrac{1}{\sqrt[3]{x}}$ . Unlike $y = \dfrac{1}{x}$ , which is its own inverse, this function maps to a different one.

Integrals and applications

The antiderivative is $-\dfrac{1}{2x^2} + C$ .

Integral	Result
$\int_1^{\infty} \dfrac{dx}{x^3}$	converges to $\dfrac{1}{2}$
$\int_0^1 \dfrac{dx}{x^3}$	diverges at the origin

In physics this shape describes quantities falling off as the cube of distance: the field of an electric or magnetic dipole¹, and the tidal force the Moon and Sun exert on the Earth².

Dipole, Wikipedia
Tidal force, Wikipedia

Graph of the Function y=1/x3y = 1/x^3y=1/x3