📚 Functions in IB Mathematics: A Comprehensive Guide | IB数学函数完全指南
Functions form the backbone of IB Mathematics, appearing in every topic from algebra to calculus. Understanding the concept of a function, its notation, and its transformations is essential for both Analysis & Approaches (AA) and Applications & Interpretation (AI) syllabi. This guide provides a thorough review of functions, covering definitions, properties, types, operations, and real-world modelling, all aligned with IB assessment objectives.
函数是IB数学的主干，从代数到微积分无处不在。掌握函数的概念、表示法和变换，对于数学分析与方法(AA)以及数学应用与解释(AI)课程都至关重要。本指南全面回顾函数知识，涵盖定义、性质、类型、运算和现实建模，紧密贴合IB考试目标。
1. Definition and Basic Concepts | 定义与基本概念
A function is a relation that associates each element from a set X, called the domain, with exactly one element in a set Y, called the codomain. If a relation has a single input producing more than one output, it cannot be a function. The familiar vertical line test on a graph quickly checks this: if any vertical line cuts the graph more than once, the relation is not a function.
函数是一种关系，它将来自集合X（定义域）的每个元素与集合Y（值域范围）中的恰好一个元素配对。若某个关系对同一个输入产生多个输出，它就不是函数。图形上的垂直线检验可快速验证：若任何垂直线与图像相交超过一次，该关系就不是函数。
We write f: X → Y, read as ‘f maps from X to Y’. For an individual element, f(x) denotes the output when x is the input. The set of all possible outputs actually obtained is called the range, which is a subset of the codomain. In IB problems, you often need to distinguish between these sets.
我们记作 f: X → Y，读作“f从X映射到Y”。对单个元素，f(x) 表示以 x 为输入时的输出。所有实际可能取得的输出构成的集合称为值域，它是陪域的子集。在IB题目中，常需区分这些集合。
2. Domain and Range | 定义域与值域
Determining the domain of a function requires examining the expression for restrictions: denominators cannot be zero, radicands of even roots must be non-negative, and arguments of logarithms must be positive. For example, the domain of f(x) = √(x – 2) is x ≥ 2 because the square root demands x – 2 ≥ 0. For g(x) = 1/(x² – 1), we exclude x = ±1, so the domain is {x ∈ ℝ | x ≠ -1, x ≠ 1}.
确定定义域需检查表达式中的所有限制条件：分母不能为零，偶次根号下的被开方数必须非负，对数的真数必须为正。例如，f(x) = √(x – 2) 的定义域是 x ≥ 2，因为平方根要求 x – 2 ≥ 0。对于 g(x) = 1/(x² – 1)，需要排除 x = ±1，定义域为 {x ∈ ℝ | x ≠ -1, x ≠ 1}。
The range is the set of all resulting y-values. For many functions, sketching the graph helps visualise the range. For f(x) = x², the range is y ≥ 0. For h(x) = 2/(x – 3), the horizontal asymptote is y = 0 so the range is y ≠ 0. Algebraic manipulation, such as solving y = f(x) for x, can also determine the range.
值域是所有y值的集合。对许多函数，画图有助于直观看到值域。f(x) = x² 的值域是 y ≥ 0。h(x) = 2/(x – 3) 的水平渐近线是 y = 0，因此值域为 y ≠ 0。通过代数变形，如将 y = f(x) 解出 x，也可确定值域。
3. Types of Functions | 函数的类型
IB covers a wide range of function families. Linear functions have the form f(x) = mx + c, producing straight lines with slope m and y-intercept c. Quadratic functions f(x) = ax² + bx + c create parabolas; their vertex and axis of symmetry are key features tested in exams.
IB包含多种函数族。线性函数形如 f(x) = mx + c，图像是斜率为 m、y截距为 c 的直线。二次函数 f(x) = ax² + bx + c 产生抛物线；顶点和对称轴是考试中的核心特征。
Polynomial functions of higher degree are examined for end behaviour and roots. Rational functions, such as f(x) = (3x+1)/(x-2), feature asymptotes and holes. Exponential functions f(x) = a·bˣ and logarithmic functions f(x) = ln x model growth and decay. Trigonometric functions sin x, cos x, and tan x are periodic and foundational for calculus and waves.
更高次的多项式函数需要研究末端走势和根。有理函数如 f(x) = (3x+1)/(x-2) 具有渐近线和可去间断点。指数函数 f(x) = a·bˣ 和对数函数 f(x) = ln x 模拟增长与衰减。三角函数 sin x、cos x 和 tan x 是周期函数，是微积分和波动的基础。
Additionally, functions can be classified as algebraic or transcendental. Understanding the parent function of each type helps in predicting the effects of transformations, which is a heavily examined topic in IB Paper 1 and Paper 2.
此外，函数还可分为代数函数和超越函数。掌握每一类母函数有助于预判变换的效果，这是IB试卷1和试卷2中频繁考查的内容。
4. Function Notation and Evaluation | 函数表示法与求值
The notation f(x) does not mean multiplication; it indicates the output of function f for input x. Evaluating f(a) simply substitutes a into the function’s rule. For f(x) = x² – 3x + 2, f(5) = 5² – 3·5 + 2 = 12. This direct substitution is a basic skill tested throughout the syllabus.
记号 f(x) 并不表示乘法，而是代表函数 f 在输入 x 下的输出。计算 f(a) 只需将 a 代入函数表达式。对 f(x) = x² – 3x + 2，f(5) = 5² – 3·5 + 2 = 12。这种直接代值是贯穿整个课程的基本技能。
IB questions frequently connect evaluation to graphs, tables, or composite functions. For example, given a graph of f, you may read f(2) directly. When the function is defined by a mapping diagram or a table, you interpret the ordered pairs accordingly. Always pay attention to notation such as (f + g)(x) = f(x) + g(x) and (f·g)(x) = f(x)·g(x).
IB题目常将求值与图像、表格或复合函数结合。例如，给定 f 的图像，可以直接读取 f(2) 的值。当函数由映射图或表格定义时，需要依据有序对来理解。务必注意 (f + g)(x) = f(x) + g(x) 以及 (f·g)(x) = f(x)·g(x) 等运算表示法。
5. Properties of Functions | 函数的性质
A function is even if f(-x) = f(x) for all x in the domain; its graph is symmetric about the y-axis. Typical examples are f(x) = x², f(x) = cos x. A function is odd if f(-x) = -f(x); its graph shows rotational symmetry of 180° about the origin, e.g., f(x) = x³, f(x) = sin x.
若对定义域内所有 x 都有 f(-x) = f(x)，则该函数为偶函数；其图像关于y轴对称。典型例子是 f(x) = x²、f(x) = cos x。若满足 f(-x) = -f(x)，则为奇函数，图像关于原点180°旋转对称，例如 f(x) = x³、f(x) = sin x。
Monotonicity describes whether a function is always increasing or always decreasing on an interval. A function is increasing if a < b implies f(a) < f(b). These properties are essential when analysing behaviour of graphs and when proving inequalities. In IB, you might be asked to determine intervals of increase/decrease using derivatives or by inspecting the graph.
单调性描述函数在某个区间上是否始终递增或递减。若 a < b 蕴含 f(a) < f(b)，则函数递增。这些性质在分析图形走势和证明不等式时非常关键。在IB中，可能需要利用导数或观察图像来确定递增/递减区间。
Another important property is one-to-one (injective): a function where each output comes from exactly one input. This occurs if every horizontal line intersects the graph at most once – the horizontal line test. Only one-to-one functions have inverses that are also functions.
另一个重要性质是单射（一一映射）：每个输出都来自恰好一个输入。若每条水平线与图像最多相交一次，即通过水平线检验，则该函数是单射。只有单射函数才有函数形式的反函数。
6. Composite Functions | 复合函数
A composite function is formed by applying one function to the output of another. The notation (f∘g)(x) = f(g(x)) means first apply g to x, then apply f to the result. The order is crucial: f(g(x)) is generally not the same as g(f(x)). Vocabulary: g is the inner function, f is the outer function.
复合函数是将一个函数作用于另一个函数的输出而形成的。记号 (f∘g)(x) = f(g(x)) 表示先将 g 作用于 x，再将 f 作用于所得结果。顺序至关重要：f(g(x)) 通常不等于 g(f(x))。术语上，g 为内层函数，f 为外层函数。
The domain of f∘g is a subset of the domain of g: it consists of all x in the domain of g such that g(x) is in the domain of f. For instance, if f(x) = √x and g(x) = x – 3, then (f∘g)(x) = √(x – 3). The domain requires x – 3 ≥ 0, so x ≥ 3, even though g(x) is defined for all real numbers.
f∘g 的定义域是 g 定义域的子集：它包含 g 定义域中所有使得 g(x) 落在 f 定义域内的 x。例如，若 f(x) = √x，g(x) = x – 3，则 (f∘g)(x) = √(x – 3)。定义域要求 x – 3 ≥ 0，即 x ≥ 3，尽管 g(x) 原本对所有实数都有定义。
IB exams frequently ask for explicit expressions for f(g(x)) and then require you to find its domain or range. Practise separating the steps: compute the inner function first, then simplify the outer, and finally check for domain restrictions introduced at any stage.
IB考试常要求求出 f(g(x)) 的具体表达式，然后确定其定义域或值域。练习时按步骤操作：先计算内层函数，再简化外层，最后检查任何阶段引入的定义域限制。
7. Inverse Functions | 反函数
An inverse function f⁻¹ reverses the effect of f. If f(a) = b, then f⁻¹(b) = a. For this to define a function, f must be one-to-one. We can find the inverse by writing y = f(x), swapping x and y, and then solving for y. The result is y = f⁻¹(x).
反函数 f⁻¹ 逆转 f 的效果。若 f(a) = b，则 f⁻¹(b) = a。要使反函数仍是函数，f 必须是单射。求反函数的方法为：写出 y = f(x)，交换 x 与 y，然后解出 y，所得即为 y = f⁻¹(x)。
The domain of f becomes the range of f⁻¹, and the range of f becomes the domain of f⁻¹. Graphically, the graph of y = f⁻¹(x) is the reflection of the graph of y = f(x) in the line y = x. This reflection property helps in sketching inverse graphs and checking symmetry.
f 的定义域成为 f⁻¹ 的值域，f 的值域成为 f⁻¹ 的定义域。从图像上看，y = f⁻¹(x) 是 y = f(x) 关于直线 y = x 的反射。这一反射性质有助于绘制反函数图像并检查对称性。
Not all functions have an inverse over their entire domain; sometimes we restrict the domain to make the function one-to-one. For example, f(x) = x² is not invertible on ℝ, but if we restrict the domain to x ≥ 0, then f⁻¹(x) = √x. IB questions often require stating the restricted domain and the corresponding inverse.
并非所有函数在整个定义域上都有反函数；有时需要限制定义域以使其成为单射。例如，f(x) = x² 在 ℝ 上不可逆，但若将定义域限制在 x ≥ 0，则 f⁻¹(x) = √x。IB题目常要求写出该限制定义域及相应的反函数。
8. Transformations of Functions | 函数的变换
Transformations allow us to shift, stretch, compress, and reflect graphs. The general transformed form is y = af(b(x – h)) + k, where each parameter has a distinct effect. Understanding these parameters is crucial for graph sketching and function modelling in IB.
变换让我们能够平移、伸缩、压缩和反射图像。一般变换式为 y = af(b(x – h)) + k，每个参数有不同的效果。理解这些参数对于IB中的图像绘制与函数建模至关重要。
The table below summarises the main transformations applied to y = f(x):
| Transformation | Notation | Effect |
|---|---|---|
| Vertical translation up by k | y = f(x) + k | Shifts graph k units upward (k > 0) or downward (k < 0) |
| Horizontal translation right by h | y = f(x – h) | Shifts graph h units right (h > 0) or left (h < 0) |
| Vertical stretch/compression by a | y = a f(x) | Stretches (|a|>1) or compresses (0<|a|<1) vertically; reflects over x-axis if a<0 |
| Horizontal stretch/compression by 1/b | y = f(bx) | Compresses (|b|>1) or stretches (0<|b|<1) horizontally; reflects over y-axis if b<0 |
下表总结了作用于 y = f(x) 的主要变换：
| 变换 | 表示法 | 效果 |
|---|---|---|
| 向上垂直平移 k 个单位 | y = f(x) + k | 图像向上(k > 0)或向下(k < 0)平移 k 个单位 |
| 向右水平平移 h 个单位 | y = f(x – h) | 图像向右(h > 0)或向左(h < 0)平移 h 个单位 |
| 垂直伸缩 a 倍 | y = a f(x) | 垂直拉伸(|a|>1)或压缩(0<|a|<1)；若 a<0 则关于 x 轴对称反射 |
| 水平伸缩 1/b 倍 | y = f(bx) | 水平压缩(|b|>1)或拉伸(0<|b|<1)；若 b<0 则关于 y 轴对称反射 |
When combining transformations, apply horizontal translations and stretches first (everything inside the brackets), then vertical stretches and translations. This order prevents errors in sketching.
组合变换时，先进行水平平移与伸缩（括号内的运算），再进行垂直伸缩与平移。按此顺序操作可避免绘图错误。
9. Absolute Value and Piecewise Functions | 绝对值函数与分段函数
The absolute value function is defined as |x| = x if x ≥ 0, and |x| = -x if x < 0. Its graph is V-shaped with vertex at the origin. Transformations of |x|, such as y = |x - 3| + 2, shift the vertex. When solving equations like |2x - 1| = 5, consider both cases: 2x - 1 = 5 and 2x - 1 = -5.
绝对值函数定义为：当 x ≥ 0 时 |x| = x；当 x < 0 时 |x| = -x。其图像呈V形，顶点在原点。|x| 的变换如 y = |x - 3| + 2 会平移顶点。解方程 |2x - 1| = 5 时，需考虑两种情况：2x - 1 = 5 和 2x - 1 = -5。
Piecewise functions are defined by different rules on different intervals. For example, a function might be defined as f(x) = x² for x ≤ 1 and f(x) = 2 – x for x > 1. To evaluate, first determine which interval contains the input, then use the corresponding rule. Graphing piecewise functions requires plotting each sub-function on its specified interval, paying attention to open or closed endpoints.
分段函数在不同区间上由不同法则定义。例如，一个函数可定义为：x ≤ 1 时 f(x) = x²，x > 1 时 f(x) = 2 – x。求值时，先判断输入落在哪一区间，再使用相应表达式。绘制分段函数图像需在每个指定区间上分别绘制子函数，并注意端点的开闭。
IB frequently combines absolute value with piecewise definitions, asking for the function’s expression as a hybrid function or for its graph. Mastery of interval notation and inequality skills is indispensable here.
IB常将绝对值与分段定义结合，要求写出混合形式的分段表达式或画出图像。掌握区间标记法和不等式技巧在此不可或缺。
10. Applications of Functions | 函数的应用
Functions model real-world phenomena: population growth can be modelled with P(t) = P₀e^(kt); antibiotic decay follows A(t) = A₀·2^(-t/h). Linear functions describe situations with constant rate of change, while quadratics model projectile motion and profit optimisation.
函数可以模拟现实世界现象：人口增长可用 P(t) = P₀e^(kt) 建模；抗生素衰减遵循 A(t) = A₀·2^(-t/h)。线性函数描述变化率恒定的场景，二次函数则模拟抛体运动和利润优化问题。
IB Internal Assessments (IA) often involve choosing a suitable function family, fitting parameters, and commenting on limitations. For example, a logistic function may better model disease spread with a limiting capacity K: N(t) = K / (1 + ce^(-rt)).
IB内部评估(IA)常常涉及选择合适的函数族、拟合参数并评论局限性。例如，逻辑斯谛函数可以更好地模拟具有容量限制 K 的疾病传播：N(t) = K / (1 + ce^(-rt))。
In the AI syllabus, students work with sinusoidal functions to model tidal data or temperature cycles: T(t) = A sin(B(t – C)) + D. In AA, functions are pushed further into calculus contexts. Both paths require linking a function’s parameters to physical meaning.
在AI课程中，学生运用正弦函数模拟潮汐数据或温度周期：T(t) = A sin(B(t – C)) + D。在AA课程中，函数更深入地融入微积分情境。两条路径都要求将函数参数与物理意义联系起来。
When solving application problems, clearly define variables, state the domain relevant to the context, and interpret your results. For instance, a negative time output might be mathematically valid but irrelevant in a real scenario, so you restrict accordingly.
解决应用题时，需明确定义变量，写出与情境相关的定义域，并解释结果。例如，负的时间输出在数学上可能有效，但在实际情境中无意义，因此需要相应限制。
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