📚 Functions | 函数
A function lies at the heart of nearly every topic in the IB Mathematics: Core syllabus. It allows us to model relationships, analyse real-world data and solve equations by linking each input to exactly one output. In this revision guide, we explore the essential concepts you must master: definitions, domain and range, notation, composite and inverse functions, graphical transformations, parity, and common elementary functions. Whether you are studying at SL or HL, a solid grasp of functions will sharpen your problem-solving skills and prepare you for questions across algebra, calculus and beyond.
函数是 IB 数学核心课程中几乎所有主题的核心。它使我们能够通过将每个输入与唯一输出联系起来，对关系进行建模、分析现实数据并求解方程。在本复习指南中，我们将探讨必须掌握的基本概念：定义、定义域与值域、表示法、复合函数与反函数、图形变换、奇偶性以及常见的初等函数。无论你学习的是 SL 还是 HL，牢固掌握函数都会提高你的解题能力，并为代数、微积分等领域的题目做好准备。
1. What is a Function? | 什么是函数？
et X (the domain) exactly one element y in a set Y (the codomain). We write y = f(x). This 'one output per input' reas x² + y² = 1 does not, because for a given x-value there may be two possible y-values.
函数 f 是一种规则，它将集合 X（定义域）中的每个元素 x 分配给集合 Y（上域）中唯一一个元素 y，记作 y = f(x)。这种“一个输入对应一个输出”的条件通常用垂直线检验：如果任何垂直线与图形相交多于一次，则该关系不是函数。例如，方程 y = x² 定义了一个函数，而 x² + y² = 1 则不是，因为对于一个给定的 x 值，可能有两个 y 值与之对应。
2. Function Notation and Evaluation | 函数表示法与求值
3. This notation is extremely useful when dealing with composite or inverse functions and when applying transformations. It also emphasises that the output depends solely on the input variable.
函数表示法提供了一种描述规则的简洁方式。如果 f(x) = 2x² − 3x + 1，那么 f(2) 意味着将 x 替换为 2：f(2) = 2(2)² − 3(2) + 1 = 8 − 6 + 1 = 3。在处理复合函数或反函数以及应用变换时，这种表示法非常有用。它还强调了输出完全取决于输入变量。
3. Domain and Range | 定义域与值域
nction is defined. The range is the set of all output values actually produced. For f(x) = √x − 1, the expression under the square root must be non‑negative, so x − 1 ≥ 0, giving the domain x ∈ [1, ∞). The smallest output is √0 = 0, and as x increases, √x−1 grows without bound, so the range is y ∈ ors that cannot be zero or logarithms that require positive arguments often dictate the domain.
函数的定义域是函数有定义的所有可能输入值的集合。值域是实际产生的所有输出值的集合。对于 f(x) = √x − 1，平方根下的表达式必须非负，因此 x − 1 ≥ 0，定义域为 x ∈ [1, ∞)。最小输出为 √0 = 0，随着 x 增大，√x−1 无限增大，因此值域为 y ∈ [0, ∞)。分母不能为零、对数要求正的真数等限制往往决定了定义域。
4. Composite Functions | 复合函数
A composite function applies one function to the result of another. The notation (f ∘ f(x) = 2x + 1 and g(x) = x², then f(g(x)) = 2(x²) + 1 = 2x² + 1. Note that the domain of f ∘ enerally commutative: g(f(x)) = (2x + 1)² = 4x² + 4x + 1, which is different from f(g(x)).
复合函数是将一个函数作用于另一个函数的结果。记法 (f ∘ g)(x) 或 f(g(x)) 表示：先计算 g(x)，然后将该输出代入 f。例如，若 f(x) = 2x + 1，g(x) = x²，则 f(g(x)) = 2(x²) + 1 = 2x² + 1。请注意，f ∘ g 的定义域包含 g 的定义域中所有使得 g(x) 落在 f 的定义域内的 x。复合运算一般不满足交换律：g(f(x)) = (2x + 1)² = 4x² + 4x + 1，这与 f(g(x)) 不同。
5. Inverse Functions | 反函数
An inverse function, denoted f⁻¹(x), 'undoes' the action of f. For f⁻¹ to exist as a function, f must be one‑to‑ = (x + 3)/(x − 2). The domain of f⁻¹ is the range of f, and vice versa. Graphically, f and f⁻¹ are reflections of each other across the line y = x.
反函数记作 f⁻¹(x)，它“撤销” f 的作用。要使 f⁻¹ 作为函数存在，f 必须是一对一的（单射），这可通过水平线检验验证。代数求反函数时，设 y = f(x)，交换 x 和 y，然后解出 y。例如，若 f(x) = (2x + 3)/(x − 1)，令 y = (2x + 3)/(x − 1)，交换得 x = (2y + 3)/(y − 1)，然后求解： = 2y + 3 → xy − x = 2y + 3 → xy − 2y = x + 3 → y(x − 2) = x + 3 → f⁻¹(x) = (x + 3)/(x − 2)。f⁻¹ 的定义域是 f 的值域，反之亦然。在图形上，f 和 f⁻¹ 关于直线 y = x 对称。
6. Even and Odd Functions | 奇函数与偶函数
(−x) = f(x) for all x in its domain. Graphically, even functions are symmetric about the y‑axis. Examples include f(x) = x², f(x) = cos x and f(x) = |x| r even nor odd. Recognising parity can simplify problems, such as when evaluating definite integrals over symmetric intervals or when analysing the behaviour of a function.
如果对于定义域内所有 x 均有 f(−x) = f(x)，则函数为偶函数。从图形上看，偶函数关于 y 轴对称，例如 f(x) = x²、f(x) = cos x 和 f(x) = |x|。如果 f(−x) = −f(x)，则函数为奇函数，图形关于原点对称，例如 f(x) = x³、f(x) = sin x 和 f(x) = 1/x。许多函数既非奇函数也非偶函数。识别奇偶性可以简化问题，比如在对称区间上计算定积分或分析函数行为时。
7. Transformations of Functions | 函数的变换
Transformations allow us to sketch new functions from a parent graph. Key transformations include:
- Vertical translation: y = f(x) + k moves the graph up (k > 0) or down (k < 0).
- Horizontal translation: y = f(x − h) shifts the graph right (h > 0) or left (h < 0).
- Vertical stretch/compression: y = a f(x) stretches vertically by factor |a|; if a is negative, the graph also reflects in the x‑axis.
- Horizontal stretch/compression: y = f(bx) compresses horizontally by factor 1/|b| (if |b| > 1) or stretches (if 0 < |b| < 1); negative b reflects in the y‑axis.
- Reflections: y = −f(x) reflects across the x‑axis; y = f(−x) reflects across the y‑axis.
变换使我们能够从母图出发绘制新函数的图形。主要变换包括：
- 垂直平移：y = f(x) + k 将图形向上 (k > 0) 或向下 (k < 0) 移动。
- 水平平移：y = f(x − h) 将图形向右 (h > 0) 或向左 (h < 0) 移动。
- 垂直拉伸/压缩：y = a f(x) 以因子 |a| 垂直拉伸；若 a 为负，图形还关于 x 轴反射。
- 水平拉伸/压缩：y = f(bx) 以因子 1/|b| 水平压缩（若 |b| > 1）或拉伸（若 0 < |b| < 1）；b 为负时关于 y 轴反射。
- 反射：y = −f(x) 关于 x 轴反射；y = f(−x) 关于 y 轴反射。
8. Basic Elementary Functions | 基本初等函数
IB Mathematics expects you to be familiar with the graphs and properties of several parent functions. These include:
- Linear: f(x) = mx + c; constant rate of change.
- Quadratic: f(x) = ax² + bx + c (a ≠ 0); parabola with vertex and axis of symmetry.
- Cubic: f(x) = ax³ + bx² + cx + d; up to two turning points.
- Reciprocal: f(x) = 1/x; vertical asymptote at x = 0, horizontal asymptote at y = 0.
- Absolute value: f(x) = |x|; V‑shaped graph, even function.
- Exponential: f(x) = aˣ (a > 0, a ≠ 1); horizontal asymptote y = 0, growth or decay.
- Logarithmic: f(x) = logₐ x (a > 0, a ≠ 1); inverse of exponential, vertical asymptote x = 0.
- Trigonometric: sin x, cos x, tan x; periodic behaviour, specific domains for inverses.
IB 数学要求你熟悉若干母函数的图形和性质。这些包括：
- 线性函数：f(x) = mx + c；变化率恒定。
- 二次函数：f(x) = ax² + bx + c (a ≠ 0)；具有顶点和对称轴的抛物线。
- 三次函数：f(x) = ax³ + bx² + cx + d；最多两个转折点。
- 倒数函数：f(x) = 1/x；垂直渐近线 x = 0，水平渐近线 y = 0。
- 绝对值函数：f(x) = |x|；V 形图，偶函数。
- 指数函数：f(x) = aˣ (a > 0, a ≠ 1)；水平渐近线 y = 0，增长或衰减。
- 对数函数：f(x) = logₐ x (a > 0, a ≠ 1)；指数函数的反函数，垂直渐近线 x = 0。
- 三角函数：sin x、cos x、tan x；呈现周期性，其反函数有特定的定义域。
9. Solving Equations Involving Functions | 涉及函数的方程求解
onsider f(x) = x² − 3 and g(x) = 2x + 1. To solve f(g(x)) = 1, substitute: (2x + 1)² − 3 = 1 → (2x + 1)² = 4 → 2x + 1 = ±2 → x = 0.5 or x = −1.5. When absolute values appear, split into cases. Always check that solutions lie within the allowed domain.
涉及函数的方程通常需要代数操作、图形分析或两者结合。例如，求解 f(x) = g(x) 可进行代数求解或找到两图形交点。对于复合函数，可以通过先确定内层表达式来求解 f(g(x)) = c。考虑 f(x) = x² − 3，g(x) = 2x + 1。求解 f(g(x)) = 1：代入得 (2x + 1)² − 3 = 1 → (2x + 1)² = 4 → 2x + 1 = ±2 → x = 0.5 或 x = −1.5。当出现绝对值时，要分情况讨论。务必检查解是否落在允许的定义域内。
10. Graphical Interpretation and Key Features | 图形解读与关键特征
often ask you to sketch or interpret graphs. You should be able to identify intercepts (x‑ and y‑ used to solve inequalities: for f(x) > g(x), look for where the graph of f lies above that of g. Furthermore, understanding how the derivative relates to a function is touched upon in calculus, but the core function topics build the necessary intuition for later studies.
IB 考题经常要求你绘制或解读图形。你应能够识别截距（x 和 y）、转折点、渐近线、递增和递减区间以及对称性。y = f(x) 的图形也可用于求解不等式：对于 f(x) > g(x)，观察 f 的图形在 g 图形上方的部分。此外，尽管在微积分中会涉及导数与函数的关系，但核心函数专题为后续学习建立了必要的直观认识。
11. Piecewise and Absolute Value Functions | 分段函数与绝对值函数
Piecewise functions are defined by different rules on different intervals. A classic example is the absolute value function:
f(x) = |x| = { x, if x ≥ 0; −x, if x < 0 }
ain, range and solving equations. To sketch a piecewise function, mark the boundaries and plot the relevant segment on each interval. Absolute value equations like |2x − 5| ys verify solutions satisfy the condition for that piece. These functions frequently appear in modelling and inequality questions.分段函数在不同区间上由不同的规则定义。一个经典例子是绝对值函数：
f(x) = |x| = { x, 若 x ≥ 0; −x, 若 x < 0 }
处理此类函数时，要分别考虑每一段对于定义域、值域和方程求解的影响。绘制分段函数图形时，标记边界点，并在每个区间上画出相应部分。绝对值方程如 |2x − 5| = 3 可化为两个线性方程：2x − 5 = 3 或 2x − 5 = −3，得 x = 4 或 x = 1。务必验证解满足该分段的条件。这些函数经常出现在建模和不等式问题中。Published by TutorHao | Mathematics Revision Series | aleveler.com
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