Graph of the Function $y = x|x|$

$y = x|x|$ equals $x^2$ for $x \geq 0$ and $-x^2$ for $x < 0$. It keeps the right half of a parabola and reflects the left half through the origin, turning the even function $x^2$ into an odd one. It is the simplest well-known example of a function that can be differentiated once but not twice.

Domain and symmetry

Both the domain and the range are all of the real numbers. Since $f(-x) = (-x)|-x| = -x|x| = -f(x)$, the function is odd, with rotational symmetry about the origin. It is positive for $x > 0$ and negative for $x < 0$, and the origin is its only zero.

Monotonicity

The derivative is $2x$ for $x > 0$ and $-2x$ for $x < 0$, which can be written together as a single formula.

At the origin the one-sided derivatives are both $0$ and therefore agree, so $y'(0) = 0$ is well defined. The derivative thus exists on the whole real line and is continuous. Since $y' \geq 0$ the function increases throughout and has no extrema.

Differentiable once but not twice

The derivative $y' = 2|x|$ is continuous, but being an absolute value it is not differentiable at the origin. The second derivative is $2$ for $x > 0$ and $-2$ for $x < 0$, values that disagree at $0$. The function is therefore continuously differentiable once but not twice. In a single short formula it shows that smoothness comes in degrees.

Concavity

From the sign of the second derivative, the curve is concave up for $x > 0$ and concave down for $x < 0$, so the concavity reverses at the origin, making it an inflection point. The tangent there is horizontal and yet there is no extremum, just as for $y = x^3$.

The difference is that $x^3$ can be differentiated any number of times, whereas this function stumbles on the second derivative. The two graphs meet at $x = 0$ and $x = \pm 1$, with $x|x|$ bulging further out near the origin.

Inverse function

On the side where $y \geq 0$, solving $y = x^2$ gives $x = \sqrt{y}$; where $y < 0$, solving $y = -x^2$ gives $x = -\sqrt{-y}$. Together the inverse is written as follows.

This is the so-called signed square root, the even function $\sqrt{|x|}$ rebuilt as an odd one. It is constructed in exactly the same way as the original function.

Integral

The antiderivative is $\dfrac{x^2|x|}{3}$, which equals $\dfrac{x^3}{3}$ for $x > 0$ and $-\dfrac{x^3}{3}$ for $x < 0$; differentiating on either interval returns $x|x|$. Being odd, the function integrates to $0$ over any interval symmetric about the origin.

Applications

Air resistance proportional to the square of the speed is written, once direction is taken into account, as proportional to $-v|v|$. Writing simply $-v^2$ would leave the drag pointing backwards even when the body moves backwards; multiplying by $|v|$ keeps the magnitude quadratic while making the direction always oppose the motion. A plot of drag against velocity is exactly this curve. In numerical analysis it also serves as a test function for checking the order of smoothness a method requires.