📚 IB Mathematics: Core Knowledge of Functions | IB数学：函数核心知识梳理
Functions form the backbone of IB Mathematics, appearing in every paper from Paper 1 to Paper 3, whether you are studying Analysis and Approaches (AA) or Applications and Interpretation (AI). This guide consolidates the core knowledge you need to master functions confidently.
函数是 IB 数学的基石，无论你是学习分析与方法（AA）还是应用与解释（AI），函数都贯穿 Paper 1 至 Paper 3 的所有考卷。本文为你系统梳理函数的核心知识，助你自信应对考试。
1. Domain and Range | 定义域与值域
The domain of a function is the set of all possible input values (x) for which the function is defined. The range is the set of all possible output values (f(x)) that result from those inputs.
函数的定义域是所有可能的输入值（x）的集合，函数在这些输入值上有定义；值域则是这些输入所对应的所有输出值（f(x)）的集合。
For polynomial functions, the domain is all real numbers, denoted ℝ.
对于多项式函数，定义域为全体实数，记作 ℝ。
unctions like f(x) = 1⁄(x − 2), exclude values that make the denominator zero: Domain = {x ∈ ℝ | x ≠ 2}.
对于有理函数如 f(x) = 1⁄(x − 2)，需排除使分母为零的值：定义域 = {x ∈ ℝ | x ≠ 2}。
For square root functions like f(x) = √x − 3, the expression inside must be non-negative: Domain = {x ∈ ℝ | x ≥ 3}.
对于平方根函数如 f(x) = √x − 3，根号内的表达式必须非负：定义域 = {x ∈ ℝ | x ≥ 3}。
Always state domain and range using interval notation or set-builder notation in your exam answers.
考试作答时，务必使用区间记号或集合构造记号来书写定义域和值域。
2. Function Notation and Evaluation | 函数记号与求值
Function notation f(x) tells us the rule applied to the input x. Understanding how to evaluate, substitute, and interpret this notation is essential.
函数记号 f(x) 表示作用于输入 x 的运算规则。理解如何求值、代入和解读这一记号至关重要。
If f(x) = 2x² − 3x + 1, then f(2) = 2(2)² − 3(2) + 1 = 8 − 6 + 1 = 3.
若 f(x) = 2x² − 3x + 1，则 f(2) = 2(2)² − 3(2) + 1 = 8 − 6 + 1 = 3。
tituting g(x) into f. For example, if f(x) = x + 1 and g(x) = x², then f(g(2)) = f(4) = 5.
复合求值 f(g(x)) 表示将 g(x) 代入 f。例如，若 f(x) = x + 1，g(x) = x²，则 f(g(2)) = f(4) = 5。
When given a graph, read function values directly from the y-coordinate at the given x-coordinate.
当给出图像时，直接读取给定 x 坐标所对应的 y 坐标即为函数值。
f(x) = 2x² − 3x + 1 → f(2) = 3
3. Composition of Functions | 复合函数
The composition (f ∘ ns applying g first, then f. The order matters: f(g(x)) is generally not equal to g(f(x)).
复合 (f ∘ g)(x) = f(g(x)) 表示先运算 g，再运算 f。运算顺序至关重要：f(g(x)) 通常不等于 g(f(x))。
Domain of f ∘ g: x must be in the domain of g, and g(x) must be in the domain of f.
复合函数 f ∘ g 的定义域：x 必须在 g 的定义域内，且 g(x) 必须在 f 的定义域内。
Example: f(x) = √x, g(x) = x − 1. Then (f ∘ g)(x) = √x − 1, with domain x ≥ 1.
例：f(x) = √x，g(x) = x − 1。则 (f ∘ g)(x) = √x − 1，定义域为 x ≥ 1。
In IB exams, you may be asked to find f ∘ g, g ∘ f, f ∘ f, or determine one function given the composite and the other function.
IB 考试中，你可能会被要求求 f ∘ g、g ∘ f、f ∘ f，或在已知复合函数和其中一个函数时求出另一个函数。
4. Inverse Functions | 反函数
= a. A function must be one-to-one (injective) to have an inverse over its entire domain.
反函数 f⁻¹ 逆转 f 的作用。若 f(a) = b，则 f⁻¹(b) = a。函数必须是一一对应（单射）才能在其整个定义域上存在反函数。
To find the inverse algebraically: replace f(x) with y, swap x and y, then solve for y.
代数求反函数的方法：将 f(x) 换成 y，交换 x 和 y，然后解出 y。
The graph of f⁻¹ is the reflection of f across the line y = x.
f⁻¹ 的图像是 f 关于直线 y = x 的镜像反射。
Domain of f⁻¹ = Range of f, and Range of f⁻¹ = Domain of f.
f⁻¹ 的定义域 = f 的值域，f⁻¹ 的值域 = f 的定义域。
Self-inverse functions (like f(x) = 1⁄x) satisfy f(f(x)) = x.
自反函数（如 f(x) = 1⁄x）满足 f(f(x)) = x。
f(x) = 2x + 3 → f⁻¹(x) = (x − 3) ⁄ 2
5. Transformations of Graphs | 图像变换
Graph transformations allow us to sketch new functions based on a known base graph. The IB syllabus requires fluency in translations, reflections, and stretches.
图像变换使我们能够基于已知的基础图像来绘制新函数的图像。IB 教学大纲要求熟练掌握平移、反射和伸缩。
| Transformation 变换 | Function 函数 | Effect 效果 |
| Vertical translation 垂直平移 | f(x) + k | Shift up k units (k > 0) 向上平移 k 个单位 |
| Horizontal translation 水平平移 | f(x − h) | Shift right h units (h > 0) 向右平移 h 个单位 |
| Reflection in x-axis 关于 x 轴反射 | −f(x) | Flip vertically 上下翻转 |
| Reflection in y-axis 关于 y 轴反射 | f(−x) | Flip horizontally 左右翻转 |
| Vertical stretch 垂直伸缩 | a·f(x) | Stretch by factor a vertically 垂直拉伸 a 倍 |
| Horizontal stretch 水平伸缩 | f(bx) | Compress horizontally by factor 1⁄b 水平压缩 1⁄b 倍 |
y = 2f(x − 1) + 3: shift right 1, stretch vertically by 2, shift up 3.
y = 2f(x − 1) + 3：先向右平移 1，垂直拉伸 2 倍，再向上平移 3。
6. Quadratic Functions | 二次函数
are the most frequently tested function type in IB Mathematics. Master the three forms and their conversions.
二次函数 f(x) = ax² + bx + c 是 IB 数学中考查频率最高的函数类型。务必掌握三种形式及其相互转换。
| Form 形式 | Expression 表达式 | Key Feature 关键特征 |
| General 一般式 | ax² + bx + c | y-intercept at (0, c) y 轴截距为 (0, c) |
| Vertex 顶点式 | a(x − h)² + k | Vertex at (h, k) 顶点为 (h, k) |
| Factored 因式式 | a(x − p)(x − q) | x-intercepts at p and q x 轴截距为 p 和 q |
Δ > 0 gives two distinct roots, Δ = 0 gives one repeated root, Δ < 0 gives no real roots.
判别式 Δ = b² − 4ac 决定实根的个数：Δ > 0 有两个不同实根，Δ = 0 有一个重根，Δ < 0 无实根。
The axis of symmetry is x = −b ⁄ (2a), which is also the x-coordinate of the vertex.
对称轴为 x = −b ⁄ (2a)，这也是顶点的 x 坐标。
Complete the square to convert from general to vertex form: x² + 6x + 5 = (x + 3)² − 4.
通过配方法将一般式转换为顶点式：x² + 6x + 5 = (x + 3)² − 4。
7. Exponential and Logarithmic Functions | 指数函数与对数函数
Exponential functions f(x) = a·bˣ = logₐ(x) are inverse operations. They model growth, decay, and many real-world phenomena.
指数函数 f(x) = a·bˣ 和对数函数 f(x) = logₐ(x) 互为逆运算。它们用于建模增长、衰减及许多现实世界中的现象。
Key laws of exponents: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1.
指数法则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1。
ws of logarithms: logₐ(xy) = logₐx + logₐy, logₐ(x⁄y) = logₐx − logₐy, logₐ(xⁿ) = n·logₐx.
对数法则：logₐ(xy) = logₐx + logₐy，logₐ(x⁄y) = logₐx − logₐy，logₐ(xⁿ) = n·logₐx。
base: logₐx = logᵦx ⁄ logᵦa. This is essential for solving equations with different bases.
换底公式：logₐx = logᵦx ⁄ logᵦa。这在求解不同底数的方程时至关重要。
The natural exponential eˣ and natural logarithm ln x satisfy: ln(eˣ) = x and eln x = x.
自然指数 eˣ 与自然对数 ln x 满足：ln(eˣ) = x 且 eln x = x。
logₐ(b) = c ⇔ aᶜ = b
8. Rational Functions | 有理函数
Q are polynomials. In IB, the most common rational function is f(x) = (ax + b) ⁄ (cx + d).
有理函数的形式为 f(x) = P(x) ⁄ Q(x)，其中 P 和 Q 是多项式。在 IB 中，最常见的有理函数是 f(x) = (ax + b) ⁄ (cx + d)。
Vertical asymptote: set the denominator equal to zero, x = −d ⁄ c.
垂直渐近线：令分母等于零，即 x = −d ⁄ c。
Horizontal asymptote: find the limit as x → ±∞. For equal degrees, it is y = a ⁄ c.
水平渐近线：求 x → ±∞ 时的极限。当分子分母次数相同时，为 y = a ⁄ c。
The x-intercept occurs when the numerator equals zero: x = −b ⁄ a.
x 轴截距出现在分子为零时：x = −b ⁄ a。
The y-intercept is f(0) = b ⁄ d.
y 轴截距为 f(0) = b ⁄ d。
9. Trigonometric Functions | 三角函数
x + c) + d are periodic. Understanding their parameters allows you to sketch them quickly.
三角函数 y = a·sin(bx + c) + d 和 y = a·cos(bx + c) + d 是周期函数。理解其参数可以快速绘制图像。
Amplitude = |a|; Period = 360° ⁄ |b| (degrees) or 2π ⁄ |b| (radians).
振幅 = |a|；周期 = 360° ⁄ |b|（角度制）或 2π ⁄ |b|（弧度制）。
Vertical shift = d; Phase shift = −c ⁄ b.
垂直位移 = d；相位移 = −c ⁄ b。
Key values to memorise: sin 0° = 0, sin 30° = ½, sin 45° = √2⁄2, sin 60° = √3⁄2, sin 90° = 1.
需要熟记的关键值：sin 0° = 0，sin 30° = ½，sin 45° = √2⁄2，sin 60° = √3⁄2，sin 90° = 1。
os²θ = 1 and tan θ = sin θ ⁄ cos θ are used constantly in solving trigonometric equations.
恒等式 sin²θ + cos²θ = 1 以及 tan θ = sin θ ⁄ cos θ 在解三角方程时经常使用。
10. Solving Equations Graphically and Analytically | 图像法与解析法求解方程
IB exams often ask you to solve equations both graphically (using your GDC) and analytically (by algebraic manipulation). Both methods have their place.
IB 考试经常要求你用图像法（使用图形计算器 GDC）和解析法（代数运算）两种方式来求解方程。两种方法各有其用武之地。
Graphical method: plot both sides of the equation as separate functions and find intersection points.
图像法：将方程两边分别绘制为两个函数，找出交点。
Analytical method: for exponential equations, take logarithms of both sides; for quadratic equations, factorise or use the quadratic formula.
解析法：对于指数方程，对方程两边取对数；对于二次方程，使用因式分解或求根公式。
Check for extraneous solutions when squaring both sides or working with logarithms — always verify in the original equation.
在等式两边平方或处理对数时，需检查是否存在增根——务必代回原方程验证。
x = (−b ± √b² − 4ac) ⁄ (2a)
11. Applications and Modelling | 应用与建模
Functions in IB Mathematics are not just abstract concepts — they model real-world situations. Recognising the underlying function type from a problem is a critical skill.
IB 数学中的函数不仅仅是抽象概念——它们还用于建模现实世界的情境。从问题中识别出潜在的函数类型是一项关键技能。
| Real-world context 现实情境 | Function model 函数模型 |
| Population growth 人口增长 | Exponential 指数函数 |
| Radioactive decay 放射性衰变 | Exponential decay 指数衰减 |
| Projectile motion 抛体运动 | Quadratic 二次函数 |
| Cooling / heating 冷却/加热 | Exponential with asymptote 带渐近线的指数函数 |
| Sound waves 声波 | Sinusoidal 正弦型函数 |
Always define variables clearly before modelling, and state the domain that makes sense in the real-world context.
建模前务必明确变量定义，并说明在现实情境中有意义的定义域。
Interpret the constants in your model: in P(t) = P₀·ekt, P₀ is the initial quantity and k is the growth (k > 0) or decay (k < 0) rate.
解释模型中的常数：在 P(t) = P₀·ekt 中，P₀ 是初始量，k 是增长率（k > 0）或衰减率（k < 0）。
12. Common Exam Pitfalls and Tips | 常见考试陷阱与技巧
Many IB students lose marks on functions due to small but avoidable errors. Here are the most common pitfalls and how to avoid them.
许多 IB 学生在函数部分因细小但可避免的错误而失分。以下是最常见的陷阱及规避方法。
Forgetting to state the domain when finding an inverse function — the domain of f⁻¹ must equal the range of f.
求反函数时忘记注明定义域——f⁻¹ 的定义域必须等于 f 的值域。
Confusing horizontal translations: f(x − 2) shifts right, not left. Remember: (x − h) shifts right for h > 0.
混淆水平平移方向：f(x − 2) 向右平移而非向左。记住：(x − h) 在 h > 0 时向右平移。
Using degrees instead of radians (or vice versa) in trigonometric functions without converting — always check the question's required mode.
三角函数中未转换就混用角度制与弧度制——务必检查题目所要求的单位制。
Dropping solutions when solving equations — for example, x² = 9 gives x = ±3, not just x = 3.
解方程时漏解——例如，x² = 9 的解为 x = ±3，而不仅仅是 x = 3。
Sketching graphs without labelling intercepts, asymptotes, and vertices — examiners award marks for labelled key features.
绘制图像时未标出截距、渐近线和顶点——考官会对标出的关键特征给分。
Mastering functions requires consistent practice. Work through past paper questions, memorise the key forms and transformations, and always check your domain and range.
掌握函数需要持续不断的练习。精做历年真题，牢记关键形式与变换，始终检查定义域和值域。
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