The Modulus Function | 模函数（绝对值函数）

📚 The Modulus Function | 模函数（绝对值函数）

The modulus function is one of the key topics in the Edexcel A-Level Mathematics specification. It appears in algebra, graphing, equation solving, and inequality problems. A solid understanding of |x| as a distance from zero helps students handle more demanding functions such as |f(x)| and f(|x|). This article explains definitions, graphs, solving techniques, and common exam pitfalls.

模函数是爱德思 A-Level 数学大纲中的核心主题之一。它出现在代数、图像、解方程和不等式问题中。把 |x| 理解为到零的距离，有助于学生处理更复杂的函数，如 |f(x)| 和 f(|x|)。本文讲解定义、图像、求解技巧以及常见考试陷阱。

1. Definition and Notation | 定义与符号

For any real number x, the modulus or absolute value of x is written as |x|. Geometrically, |x| is the distance from x to 0 on the real number line, so distance is never negative. Algebraically, if x is zero or positive, |x| equals x; if x is negative, |x| equals the opposite of x.

对于任意实数 x，x 的模或绝对值记作 |x|。从几何上看，|x| 是 x 到实数轴上 0 的距离，因此距离永远不会是负数。从代数上看，如果 x 为零或正数，|x| 等于 x；如果 x 为负数，|x| 等于 x 的相反数。

|x| = x for x ≥ 0, |x| = −x for x < 0

For example, |5| = 5, |−3| = 3, and |0| = 0. The notation |x| should not be confused with brackets or the determinant symbol in matrices.

例如，|5| = 5，|−3| = 3，|0| = 0。注意 |x| 的记号不要与括号或矩阵中的行列式符号混淆。

2. The Modulus as a Piecewise Function | 分段函数表示

Because the rule changes at x = 0,

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