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

$y = |x|$ is the magnitude of $x$ with its sign removed, that is, its distance from the origin on the number line. It equals $x$ for $x \geq 0$ and $-x$ for $x < 0$, and it can also be written $|x| = \sqrt{x^2}$.

Domain and range

The domain is all real numbers. Since the value is a distance it is never negative, so the range is $y \geq 0$.

• The domain is all real numbers
• The range is $y \geq 0$
• The minimum is $0$, attained at $x = 0$
• There is no inflection point

Symmetry

Since $|-x| = |x|$, the function is even and its graph is symmetric about the $y$-axis.

Shape of the graph

The graph is a V shape, formed by the part of the line $y = x$ with $x \geq 0$ joined to the part of the line $y = -x$ with $x < 0$, with the origin $(0, 0)$ as its vertex. It is made of two rays rather than a curve, so it bends nowhere and has no inflection point.

The corner at the origin

The derivative is $1$ for $x > 0$ and $-1$ for $x < 0$.

At the origin the slope from the left is $-1$ and the slope from the right is $1$; they disagree, so the function is not differentiable at $x = 0$. It is the standard example of a function that is continuous everywhere yet fails to be differentiable somewhere. Where the parabola $y = x^2$ has a smooth valley at the origin, $|x|$ has a sharp corner.

Solutions of the equation $|x| = a$

Notable points

The graph passes through $(\pm 1, 1)$, $(\pm 2, 2)$ and $(\pm 3, 3)$. The magnitude of the slope is $1$ everywhere, so the value grows at a constant rate as you move away from the origin.

Relationships with other functions

It equals $\sqrt{x^2}$, and also $\max(x, -x)$. Both $|x|$ and $x^2$ are even, but $x^2$ flattens near the origin while $|x|$ folds there with a constant slope. The distance to a point in the plane, $\sqrt{x^2 + y^2}$, is the two-dimensional counterpart of this function.

Applications

It measures the size of an error without regard to sign.

• In statistics, minimizing a sum of absolute values (the $L^1$ norm) gives a fit that is robust to outliers
• The triangle inequality $|a + b| \leq |a| + |b|$ underpins every branch of mathematics that deals with distance
• The ReLU of machine learning can be written $\max(x, 0) = \dfrac{x + |x|}{2}$, which is just the right half of the absolute value