y = \sqrt{|x|}

Graph of the Function $y = \sqrt{|x|}$

$y = \sqrt{|x|}$ is the square root of the absolute value of $x$ . Since $|x|$ is always non-negative, the radicand is never negative, so the function is defined on the whole real line.

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

It splits into $y = \sqrt{x}$ for $x \geq 0$ and $y = \sqrt{-x}$ for $x < 0$ . Because $|-x| = |x|$ , the graph is symmetric about the $y$ -axis: the right half is exactly $\sqrt{x}$ , reflected across the axis and spread to the left.

The cusp at the origin

Range of $x$	Derivative	Limit as $x \to 0$
$x > 0$	$\dfrac{1}{2\sqrt{x}}$	$+\infty$
$x < 0$	$-\dfrac{1}{2\sqrt{-x}}$	$-\infty$

Both tangents at the origin stand vertical, so the origin is a sharp cusp where the function is not differentiable¹. The absolute value $y = |x|$ has a corner there too, but this curve looks sharper still because the tangents are vertical rather than slanted.

Notable points

$x$	$\sqrt{\|x\|}$
$\pm 1$	$1$
$\pm 4$	$2$
$\pm 9$	$3$
$\pm 100$	$10$

Each time $|x|$ quadruples, $y$ doubles. Far from the origin the growth is much gentler than that of $|x|$ itself: at $x = 100$ the absolute value is $100$ while this function gives only $10$ .

Compared with a similar shape

It resembles $y = x^{2/3}$ , which also has a cusp at the origin, but near the origin $\sqrt{|x|}$ approaches $0$ more slowly.

$x$	$\sqrt{\|x\|}$	$x^{2/3}$
$0.01$	$0.1$	$\approx 0.046$
$1$	$1$	$1$
$100$	$10$	$\approx 21.5$

Near the origin the first is the larger, far away the second overtakes it.

Where it is used

As a basic composition of the square root and the absolute value, it is a favourite example for studying how to make a function even and how a cusp arises at the origin. It also appears when handling quantities related to distance or spread.

Singular point of a curve, Wikipedia