X-Axis
The x-axis is the horizontal number line on the coordinate plane, used to measure left and right position.
Definition
The x-axis is the horizontal number line of the coordinate plane, the set of all points $(x, 0)$ where $x$ is any real number, intersecting the y-axis at the origin $(0, 0)$; positive $x$ values lie to the right and negative values to the left. In function notation the x-axis holds the input values (domain) of a function, and its x-intercepts, where a graph crosses it, are also called the function's zeros or roots. In linear algebra terms, the x-axis is the span of the standard basis vector $e_1 = (1, 0)$, and a reflection across it is represented by the matrix $\text{diag}(1, -1)$.
Example
On the x-axis, the number $3$ is three spaces to the right of center and $-2$ is two spaces to the left; a point like $(5, 0)$ sits right on the x-axis. The x-intercept of a line is where it crosses the x-axis (where $y = 0$): for $y = 2x - 6$, setting $y = 0$ gives $0 = 2x - 6$, so $x = 3$ and the x-intercept is $(3, 0)$. The x-axis is the set of fixed points of the reflection matrix $\begin{bmatrix}1&0\\0&-1\end{bmatrix}$, whose eigenvalues are $+1$ (eigenvector $(1,0)$, the x-axis) and $-1$ (eigenvector $(0,1)$, the y-axis).
Key Insight
Any point that lies on the x-axis has a y-coordinate of $0$, since it represents the starting position before you go up or down. The choice of axes is a convention, not a geometric necessity: rotating the coordinate system by angle $\theta$ replaces the axes with new orthogonal directions, illustrating that the x-axis is simply a human-chosen reference direction within $\mathbb{R}^2$.