📚 Topic 2: Functions – SL and HL | 主题2：函数 – 标准水平与高级水平
Functions form the backbone of IB Mathematics, linking algebraic expressions to graphical representations and real-world models. In Topic 2, both SL and HL students explore domain, range, inverses, composites, and graph transformations, while HL candidates additionally tackle absolute value, polynomial, and rational functions with greater depth. This article unpacks every essential concept, provides clear definitions, and offers worked examples to help you master functions for your IB exams.
函数是 IB 数学的支柱，它将代数表达式与图像表示、实际模型紧密相连。在主题 2 中，SL 和 HL 学生都需要掌握定义域、值域、反函数、复合函数和图像变换，而 HL 学生还会深入学习绝对值函数、多项式函数和有理函数。本文将逐一剖析每一个核心概念，给出清晰的释义并通过示例讲解，帮助你彻底攻克 IB 函数专题。
1. What is a Function? | 什么是函数？
may intersect it at most once. Formal set notation is often used to describe mappings: f: ℝ → ℝ, f(x) = x² assigns each real number to its square.
函数是一种规则，它将定义域集合中的每一个输入值恰好对应到值域集合中的一个输出值。通常写作 f: x → y 或 y = f(x)。垂直线检验可以判断一个图像是否为函数图像：任意一条垂直线与图像最多只能有一个交点。描述映射时常使用正式的集合符号，例如 f: ℝ → ℝ, f(x) = x² 表示将每个实数映射为其平方。
In IB functions, inputs are independent variables (often x) and outputs are dependent variables (y or f(x)). The mapping can be one-to-one (injective), many-to-one, but never one-to-many. Understanding the basic definition prevents common errors when working with composite and inverse functions, where one-to-oneness becomes critical.
在 IB 函数中，输入是自变量（常为 x），输出是因变量（y 或 f(x)）。映射可以是一对一（单射）或多对一，但绝不能是一对多。透彻理解基本定义可以避免在处理复合函数和反函数时出现常见错误，因为这些场合对一一对应的要求十分严格。
2. Domain and Range | 定义域与值域
The domain of a function is the set of all allowable input values. For a real-valued function, we often need to exclude values that make the denominator zero or that produce a negative radicand under an even root. The range is the set of all output values actually attained by the function, and it can be read from a graph or found algebraically by considering the behaviour of f(x).
函数的定义域是所有允许的输入值的集合。对于实值函数，我们通常需要排除使分母为零或使偶次根号下出现负数的值。值域则是函数实际能够取到的所有输出值的集合，既可以从图像上观察得到，也可以通过分析 f(x) 的行为用代数方法求出。
| Function type / 函数类型 | Domain constraint / 定义域限制 |
|---|---|
| f(x) = 1/x | x ≠ 0 |
| g(x) = √x | x ≥ 0 |
| h(x) = logₐx | x > 0 |
| p(x) = 1/√x−2 | x > 2 |
For HL students, it is essential to express domains using set notation and to consider restrictions arising from multiple conditions combined. Finding the range of a rational function or a quadratic often requires completing the square or analysing asymptotic behaviour. Sketching graphs with a GDC helps visualise the domain and range quickly.
HL 学生必须学会用集合符号表示定义域，并考虑多重条件组合产生的限制。求有理函数或二次函数的值域往往需要配方法或分析渐近行为。借助图形计算器（GDC）绘制草图可以快速直观地看出定义域和值域。
3. Function Notation and Evaluation | 函数符号与求值
Function notation f(x) is read "f of x" stituting entire expressions is vital in the study of composite functions and difference quotients.
函数符号 f(x) 读作“f of x”，表示当输入为 x 时的输出值。求函数值时，直接将给定的输入表达式代入函数规则即可。例如，若 f(x) = 2x² − 3x + 1，则 f(a) = 2a² − 3a + 1，而 f(x + h) = 2(x + h)² − 3(x + h) + 1。整体表达式代入的技巧在学习复合函数和差商时至关重要。
ention to brackets and signs when substituting negative numbers or fractions. GDC table functions can speed up evaluation but do not replace algebraic manipulation skills.
IB 考试中常会要求计算函数在某一数值点的值、化简代数输出或解方程 f(x) = k。在代入负数或分数时要特别注意括号和符号。图形计算器的表格功能虽然可以加速求值，但不能取代代数运算能力。
4. Inverse Functions | 反函数
e versa. To find an inverse algebraically, write y = f(x), swap x and y, then solve for y.
反函数 f⁻¹ 能逆转 f 的作用：若 f(a) = b，那么 f⁻¹(b) = a。反函数要成为函数，原函数必须是一一对应的（单射）。从图像上看，f⁻¹ 的图像是 f 的图像关于直线 y = x 的对称图形。f 的定义域变成 f⁻¹ 的值域，反之亦然。用代数方法求反函数时，先写出 y = f(x)，交换 x 与 y，再解出 y。
al functions. HL candidates also work with inverses of logarithmic, exponential, and more complicated rational functions. They must be comfortable stating domain restrictions that ensure invertibility, and understanding why (f⁻¹ ∘ f)(x) = x.
对 SL 而言，反函数通常局限在线性函数、限定定义域后的二次函数（如 f(x) = x²，x ≥ 0）以及简单有理函数。HL 考生还需要处理对数函数、指数函数和更复杂的有理函数的反函数，必须能够给出保证可逆的定义域限制，并理解为何有 (f⁻¹ ∘ f)(x) = x。
5. Composite Functions | 复合函数
A composite function (f ∘ to the result of another. The order matters: f(g(x)) is generally not equal to g(f(x)). The domain of the composite function consists of all x in the domain of g such that g(x) lies in the domain of f. This nested domain condition is a frequent source of errors in exams.
复合函数 (f ∘ g)(x) = f(g(x)) 是指将一个函数作用于另一个函数的结果。顺序非常重要：通常 f(g(x)) 不等于 g(f(x))。复合函数的定义域由那些既属于 g 的定义域，又使得 g(x) 落在 f 的定义域内的 x 组成。这种嵌套定义域条件是考试中常见的错误来源。
Given f(x) = √x and g(x) = 2x − 3, then (f ∘ g)(x) = √2x − 3 5, which is the domain even though the individual domain of g is all reals. Conversely, (g ∘ f)(x) = 2√x − 3 has domain x ≥ 0. Always check domain explicitly.
给定 f(x) = √x 和 g(x) = 2x − 3，则 (f ∘ g)(x) = √2x − 3。要使该式为实数，必须满足 2x − 3 ≥ 0，即 x ≥ 1.5，这就是它的定义域，尽管 g 独自的定义域是所有实数。反之，(g ∘ f)(x) = 2√x − 3 的定义域是 x ≥ 0。始终要明确定义域。
6. Transformations of Graphs | 函数图像的变换
fts: f(x − c) moves the graph right by c when c > 0. Vertical shifts: f(x) + d moves the graph up by d. Reflections: −f(x) reflects in the x-axis, f(−x) reflects in the y-axis. Vertical stretch by factor a: a f(x). Horizontal stretch by factor 1/b: f(bx).
图像变换让标准函数能够延展出丰富多样的模型。关键的变换包括平移、对称和伸缩。对于函数 y = f(x)，变换后的图像可以写为 y = a f(b(x − c)) + d。水平平移：f(x − c) 当 c > 0 时图像向右平移 c 个单位。垂直平移：f(x) + d 图像向上平移 d 个单位。对称：−f(x) 关于 x 轴对称，f(−x) 关于 y 轴对称。垂直伸缩系数为 a：a f(x)；水平伸缩系数为 1/b：f(bx)。
The order of transformations matters. When dealing with b(x − c), apply the horizontal translation c after the horizontal stretch/compression of factor 1/b. IB problems often ask to describe the sequence mapping one graph onto another or to write the new equation after a set of transformations. Start from the parent function and apply changes step by step.
变换的顺序很关键。在处理 b(x − c) 时，要先应用水平伸缩/压缩（系数 1/b），再进行水平平移 c。IB 题目常常要求描述将一个图像映射到另一个图像的一系列变换，或者写出经过一组变换后的新方程。从母函数出发，逐步应用变换即可。
7. Quadratic Functions | 二次函数
a). Three algebraic forms are essential: standard form (ax² + bx + c), vertex form (a(x − h)² + k), and factored form (a(x − p)(x − q)). The vertex (h, k) gives the maximum or minimum value depending on the sign of a.
二次函数 f(x) = ax² + bx + c (a ≠ 0) 是 SL 和 HL 课程的核心内容。它的图像是一条抛物线，关于垂直轴 x = −b/(2a) 对称。三种关键的代数形式必须掌握：一般式 ax² + bx + c、顶点式 a(x − h)² + k 以及因式分解式 a(x − p)(x − q)。顶点 (h, k) 根据 a 的符号给出最大值或最小值。
ated real root; Δ < 0, no real roots. Completing the square is the bridge between standard form and vertex form, and it is frequently tested. In HL problems, the quadratic may appear in the context of rational functions or as part of a composite function, requiring careful analysis of the axis of symmetry and range.
判别式 Δ = b² − 4ac 决定了实根的个数。Δ > 0 时有两个不同的实根，Δ = 0 时有一个重根，Δ < 0 时没有实根。配方法是连接一般式与顶点式的桥梁，经常出现在考题中。在 HL 题目中，二次函数可能出现在有理函数或复合函数的背景中，要求学生仔细分析对称轴和值域。
8. Exponential and Logarithmic Functions | 指数函数与对数函数
Exponential functions f(x) = aˣ portant. Logarithmic functions f(x) = logₐx are the inverses of exponentials, satisfying alogₐx garithm ln x = logₑx appears frequently in calculus preparation and modelling. Graphs of aˣ and logₐx are reflections across the line y = x.
指数函数 f(x) = aˣ（a > 0 且 a ≠ 1）可以用来模拟连续增长和衰减。以 e ≈ 2.718 为底的指数函数尤其重要。对数函数 f(x) = logₐx 是指数函数的逆运算，满足 alogₐx = x。自然对数 ln x = logₑx 在微积分预备和建模中频繁出现。aˣ 与 logₐx 的图像关于直线 y = x 对称。
ₐx. Solving exponential equations often involves taking logs of both sides, while logarithmic equations may require exponentiating to cancel the log. Always check for extraneous solutions when the domain of the logarithm is restricted.
指数与对数运算法则必须烂熟于心：aᵐaⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，logₐ(xy) = logₐx + logₐy，logₐ(x/y) = logₐx − logₐy，logₐxⁿ = n logₐx。解指数方程时常要对两边取对数，而对数方程则可能需要指数化以消去对数。由于对数定义域受限制，必须检查可能带入的增根。
9. Rational Functions (HL) | 有理函数（高级水平）
(x) ≠ 0. HL students learn to identify vertical asymptotes at zeros of the denominator that do not cancel, horizontal asymptotes by comparing degrees of numerator and denominator, and oblique asymptotes when the degree of the numerator is exactly one more than that of the denominator. Holes occur when a factor cancels from the numerator and denominator.
有理函数是两个多项式的比值：R(x) = P(x)/Q(x)，Q(x) ≠ 0。HL 学生需要学会判断：分母在不约分情况下的零点给出垂直渐近线；通过比较分子与分母的次数确定水平渐近线；当分子次数比分母恰好高一次时则会出现斜渐近线。当分子与分母有公因式消去时，图像上会出现可去间断点（空洞）。
To sketch rational functions, first factorise fully, simplify common factors to locate holes, then find intercepts, asymptotes, and test signs of intervals. Domain is all real x except those making the simplified denominator zero. The range can often be found by considering the inverse function or using long division. These functions synthesise many algebraic techniques and appear frequently in HL Paper 2.
绘制有理函数图像时，应先彻底因式分解，约去公因式以确定空洞位置，然后求截距和渐近线，并检验各区间函数值的符号。定义域是除去使化简后分母为零的 x 之外的所有实数。值域通常可以通过反函数或长除法求出。有理函数综合了众多代数技巧，在 HL 试卷二中相当常见。
10. Polynomial Functions and Factorisation (HL) | 多项式函数与因式分解（高级水平）
Polynomial functions of degree n are expressions of the form P(x) = aₙxⁿ + … he Rational Root Theorem helps find potential rational zeros. Factoring higher-degree polynomials is crucial for solving polynomial equations and sketching graphs.
n 次多项式函数的形式为 P(x) = aₙxⁿ + … + a₁x + a₀。HL 学生需掌握余数定理（P(x) 除以 (x − a) 的余数为 P(a)）、因式定理（(x − a) 为因式当且仅当 P(a) = 0）以及多项式除法。有理根定理能帮助寻找可能的有理零点。完成高次多项式的因式分解对求解多项式方程和绘制图像至关重要。
When a factor (x − a) occurs m times, it gives a root of multiplicity m. The graph touches the x-axis at a if m is even, and crosses if m is odd. HL candidates should also be comfortable with sum and product of roots for quadratics and cubics, linking them to Vieta's formulas. Polynomial functions model many optimisation and geometric problems.
当因式 (x − a) 出现 m 次时，它给出了一个 m 重根。若 m 为偶数，图像在 x 轴处相切；若 m 为奇数，图像穿过 x 轴。HL 考生还应熟练应用二次和三次方程的根与系数关系（韦达定理）。多项式函数可以模拟许多优化与几何问题。
11. Absolute Value Functions (HL) | 绝对值函数（高级水平）
The absolute value function |x| returns the non-negative magnitude of x. Graphs involving |f(x)| reflect negative parts of f(x) above the x-axis, while f(|x| s often requires splitting into cases based on the sign of the expression inside the bars: |A| = B leads to A = B or A = −B, and inequalities require careful sign analysis.
绝对值函数 |x| 返回 x 的非负大小。含有 |f(x)| 的图像将 f(x) 的负值部分翻折到 x 轴上方，而 f(|x|) 则把图像右侧部分对称到左侧。解含有绝对值的方程通常需要根据绝对值内表达式的符号分情况讨论：|A| = B 转化为 A = B 或 A = −B，且不等式需要仔细分析符号。
adratics, exponentials, or rational functions, making sign charts essential. The graph y = |ax + b| is a V-shaped piecewise linear function. Understanding the algebraic definition |z| = √z² is also useful when manipulating expressions analytically.
HL 问题会将绝对值与二次函数、指数函数或有理函数结合，此时符号表格不可或缺。y = |ax + b| 的图像是 V 形的分段线性函数。理解代数定义 |z| = √z² 也有助于解析地处理表达式。
12. Even, Odd Functions and Periodic Behaviour | 奇偶函数与周期性
T. Trigonometric functions are the primary examples, but HL students may encounter periodic piecewise functions.
若对于定义域内所有 x 都有 f(−x) = f(x)，则函数为偶函数，其图像关于 y 轴对称；若 f(−x) = −f(x)，则为奇函数，其图像关于原点对称。这些性质有助于简化函数分析及后续微积分的运算。周期函数满足 f(x + T) = f(x)，周期为 T。三角函数是最主要的例子，但 HL 学生也可能遇到周期分段函数。
mposite transformations may alter parity. IB questions sometimes ask to prove a given function is even/odd using the definition, or to find the period from a graph. This reinforces algebraic and graphical reasoning simultaneously.
判断多项式函数的奇偶性可观察其幂次：仅含偶次项则为偶函数，仅含奇次项则为奇函数。复合变换可能改变奇偶性。IB 题目有时会要求用定义证明一个函数为偶函数或奇函数，或者从图像中读取周期。这能够同时强化代数推理和图像思维。
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