📚 Describing Functions | 函数性质的描述
In IB Mathematics, mastering the skill of describing functions is fundamental. When we describe a function, we systematically identify its key features — such as domain, range, intercepts, symmetry, monotonicity, periodicity, asymptotes, continuity, boundedness, end behaviour, and concavity. These characteristics allow us to construct accurate graphs, predict behaviour, and solve equations without relying solely on plotting points. This article will guide you through each property with clear explanations and examples relevant to the IB syllabus.
在 IB 数学中，掌握描述函数的技能是基础。描述函数时，我们系统地识别其关键特征，包括定义域、值域、截距、对称性、单调性、周期性、渐近线、连续性、有界性、末端行为以及凹凸性。这些性质使我们能够准确绘制图像、预测行为并解方程，而不必完全依赖描点法。本文将通过清晰的解释和贴合 IB 大纲的例子，带你逐一理解这些性质。
1. Domain and Range | 定义域与值域
The domain of a function f is the set of all possible input values (x-values) for which f(x) is defined. To find the domain from a formula, we must exclude values that lead to division by zero or negative radicands under even roots. The range is the set of all possible output values (y-values) that f(x) can produce. Determining the range often involves analysing the graph, considering asymptotes, or completing the square for quadratics.
函数的定义域是所有可能的输入值 x 的集合，使得 f(x) 有定义。从表达式求定义域时，需排除使分母为零或偶次根号内为负数的值。值域是所有可能的输出值 y 的集合。确定值域通常需要分析图像、考虑渐近线，或对二次函数进行配方。
For example, the function f(x) = 1/(x − 2) has domain ℝ \ except 0, since no value of x can make 1/(x−2) = 0. In interval notation, domain: (−∞, 2) ∪ (2, ∞), range: (−∞, 0) ∪ (0, ∞).
例如，函数 f(x) = 1/(x − 2) 的定义域为 ℝ \ {2}，因为 x = 2 使分母为零。其值域为除 0 外的所有实数，因为没有 x 能使 1/(x−2) = 0。用区间表示：定义域 (−∞, 2) ∪ (2, ∞)，值域 (−∞, 0) ∪ (0, ∞)。
2. Intercepts | 截距
by solving f(x) = 0. The y-intercept is where the graph crosses the y-axis, evaluated by f
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图