y = \dfrac{1}{x^2}

Graph of the Function $y = 1/x^2$

$y = \dfrac{1}{x^2}$ , also written $x^{-2}$ , is a rational function whose denominator is a square, so its values are always positive: the graph consists of two branches rising on either side of the $y$ -axis.

Domain and range

The domain is $x \neq 0$
The range is $y > 0$
Even function
No maximum and no minimum

Unlike $\dfrac{1}{x}$ , it never takes a negative value.

Symmetry

Because $(-x)^2 = x^2$ , we have $f(-x) = f(x)$ : the function is even and its graph is symmetric about the $y$ -axis, consisting of two mirror-image branches in the first and second quadrants.

Asymptotes

As $x \to 0$ from either side, $y \to +\infty$ , so the $y$ -axis is a vertical asymptote. As $x \to \pm\infty$ , $y \to 0$ , so the $x$ -axis is a horizontal asymptote, approached from above since the values stay positive.

Monotonicity

The derivative is $y' = -\dfrac{2}{x^3}$ . For $x < 0$ we have $x^3 < 0$ , so $y' > 0$ and the function increases; for $x > 0$ the derivative is negative and the function decreases. The curve simply climbs without bound as it approaches the origin from either side.

Rate of decay

$x$	$\dfrac{1}{x}$	$\dfrac{1}{x^2}$
$1$	$1$	$1$
$2$	$0.5$	$0.25$
$10$	$0.1$	$0.01$

Doubling $x$ cuts the value to $\dfrac{1}{4}$ , and tripling it cuts the value to $\dfrac{1}{9}$ . The decay is far faster than that of $\dfrac{1}{x}$ . The value $|x| = 1$ marks the divide: inside it the values exceed $1$ , outside they fall below.

Relationships with other functions

Differentiating $\dfrac{1}{x}$ produces $-\dfrac{1}{x^2}$ . Comparing the areas under the two curves beyond $x = 1$ is instructive.

Integral	Result
$\int_{1}^{\infty} \dfrac{dx}{x^2}$	converges to $1$
$\int_{1}^{\infty} \dfrac{dx}{x}$	diverges

A slight difference in how the tail decays decides whether the accumulated area is finite.

Applications

This is the inverse-square law that runs through physics¹.

Gravitational attraction between two masses
The Coulomb force between two charges
The brightness of a point source of light

Double the distance and the strength drops to $\dfrac{1}{4}$ . The reason is geometric, since the force or the light spreads evenly over a sphere, and the surface area of a sphere grows with the square of its radius.

Inverse-square law, Wikipedia

Graph of the Function y=1/x2y = 1/x^2y=1/x2