The square function

Definition

The square function is defined on ℝ by f(x) = x². It maps each real number x to the number x² = x × x.

Variations

The square function is not monotonic over the entire real line:

• on the interval ]–∞; 0], it is decreasing
• on the interval [0; +∞[, it is increasing

x	-2	-1	0	1	2
x²	4	1	0	1	4

Graph

The graph of f(x) = x² is an upward-facing parabola with its vertex at the origin (0, 0). It is symmetric about the y-axis, as f(-x) = f(x) for all x: we say that the square function is even.

Example of use

To compare f(-3) and f(2): f(-3) = 9 and f(2) = 4, so f(-3) > f(2), even though -3 < 2. This clearly illustrates that the square function is not increasing on the entire real line.

Common pitfall

Note: x² ≥ 0 always holds, but x² ≤ y² does not imply x ≤ y. For example, (-5)² = 25 > 2² = 4, even though -5 < 2. You must always consider the sign of x before comparing two squares.