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 domain is all real numbers and the range is $y \geq 0$. It splits into $y = \sqrt{x}$ for $x \geq 0$ and $y = \sqrt{-x}$ for $x < 0$.

Because $|-x| = |x|$, we have $\sqrt{|-x|} = \sqrt{|x|}$, so it is an even function, symmetric about the $y$-axis. Its right half is exactly $\sqrt{x}$, reflected across the $y$-axis and spread to the left.

For $x > 0$, $y' = \dfrac{1}{2\sqrt{x}}$; for $x < 0$, $y' = -\dfrac{1}{2\sqrt{-x}}$. As $x \to 0^{+}$, $y' \to +\infty$, and as $x \to 0^{-}$, $y' \to -\infty$, so both tangents at the origin stand vertical. The origin is therefore a sharp cusp where the function is not differentiable.

It decreases for $x < 0$ and increases for $x > 0$, with minimum value $0$ at the origin $(0, 0)$. Overall it resembles a bird spreading its wings, both sides rising gently. Concretely, $y = 1$ at $x = \pm 1$, $y = 2$ at $x = \pm 4$, and $y = 3$ at $x = \pm 9$: each time $|x|$ quadruples, $y$ doubles.

It resembles $y = x^{2/3}$, which also has a cusp at the origin, but near the origin $\sqrt{|x|}$ approaches $0$ more slowly. For instance, at $x = 0.01$, $\sqrt{|x|} = 0.1$ whereas $x^{2/3} \approx 0.046$.

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