Core Mathematics AA Functions 规划

## IB AA 函数定义域 复合反函数 图像变换 *Functions form the backbone of IB Mathematics: Analysis and Approaches. 函数是IB数学AA课程的核心支柱。 Whether you are aiming for a 7 in SL or gearing up for the HL exam, a rock solid understanding of functions is non-negotiable. This comprehensive bilingual guide walks you through every essential concept, from the foundational definition of a function to the trickiest transformation problems, with study strategies, common pitfalls, and exam focused tips woven throughout.* --- ### 1. What Is a Function? 什么是函数？ A **function** is a rule that assigns each input exactly one output. 函数是一种映射规则，每个输入值对应唯一一个输出值。 Formally, if f is a function from set A to set B, then for every x in A, there exists exactly one y in B such that f(x) = y. In IB notation, you will see three equivalent representations of a function: 三种等价表示： - **Mapping diagram** 映射图: visually shows arrows from domain elements to range elements - **Set of ordered pairs** 有序对集合: {(1, 2), (2, 4), (3, 6)} - **Algebraic rule** 代数表达式: f(x) = 2x, x ∈ ℝ The **vertical line test** is your quickest check: if any vertical line intersects the graph at more than one point, the relation is NOT a function. 垂直线检验：如果任何一条垂直线与图像的交点多于一个，则该关系不是函数。 > 💡 **Exam Tip 考试技巧**: IB examiners love asking you to distinguish between a relation and a function. Always perform the vertical line test mentally before answering. --- ### 2. Domain and Range 定义域与值域 The **domain** of a function is the set of all allowable input values (x values). The **range** is the set of all possible output values (y values). 定义域是函数所有可能的输入值的集合，值域是所有可能的输出值的集合。

🔍 Finding Domain 求定义域

Denominator ≠ 0: f(x) = 1/(x-2) domain: x ≠ 2
Square root ≥ 0: f(x) = √(x+3) domain: x ≥ -3
Log argument > 0: f(x) = ln(x-1) domain: x > 1

📐 Finding Range 求值域

Quadratic (vertex form): find y-coordinate of vertex
Rational: find horizontal asymptote as excluded value
Exponential: y > 0 if base > 0
Trigonometric: [-1, 1] for sin and cos

**Common IB trap**: When a function involves both a square root AND a denominator, you must satisfy BOTH restrictions simultaneously. 当函数同时包含根号和分母时，必须同时满足两个限制条件。 Example: f(x) = √(x+1) / (x-2) - From square root: x + 1 ≥ 0 so x ≥ -1 - From denominator: x - 2 ≠ 0 so x ≠ 2 - Domain: x ∈ [-1, 2) ∪ (2, ∞) **How IB frames domain questions 考试出题方式**: You may be asked to find the "maximal domain" (largest possible set of real numbers for which the function is defined) or "restrict the domain" so that a function becomes invertible. For HL, expect questions involving the intersection of multiple restriction conditions from logs, roots, and denominators simultaneously. --- ### 3. Composite Functions 复合函数 A **composite function** (f ∘ g)(x) = f(g(x)) means you apply g first, then feed the result into f. 复合函数是将一个函数的输出作为另一个函数的输入。 The critical rule: **the range of the inner function g must overlap with the domain of the outer function f**. 关键规则：内层函数的值域必须与外层函数的定义域有交集。

⚠️ Domain of Composite 复合定义域

To find domain of f(g(x)):
1. Find x values where g(x) is defined
2. Find x values where g(x) falls in domain of f
3. Take the INTERSECTION of both sets

🧪 Worked Example 例题

f(x) = √x, g(x) = x-4
(f∘g)(x) = √(x-4)
g defined for all x ∈ ℝ
g(x) must be ≥ 0 for f: x-4 ≥ 0
Domain: x ≥ 4

**HL Extension 高阶拓展**: For HL, you need to handle piecewise composite functions where f and g are defined differently on different intervals. You must evaluate the composition piece by piece, checking which piece of the inner function's output maps into which piece of the outer function's domain. > 💡 **Exam Tip 考试技巧**: IB often asks: "Does (f ∘ g)(x) = (g ∘ f)(x)?" The answer is almost always NO. Function composition is NOT commutative. 函数复合运算一般不满足交换律。 To disprove equality, find one counterexample value of x. --- ### 4. Inverse Functions 反函数 An **inverse function** f⁻¹(x) reverses the effect of f(x). If f(a) = b, then f⁻¹(b) = a. 反函数逆转原函数的作用。 **Geometric interpretation 几何解释**: The graph of f⁻¹ is the reflection of the graph of f across the line y = x. 反函数的图像是原函数关于直线 y = x 的镜像。 For a function to have an inverse, it must be **one-to-one (injective)**. A function is injective if f(a) = f(b) implies a = b. Use the **horizontal line test**: if any horizontal line intersects the graph more than once, the function is NOT invertible on its entire domain. 水平线检验：如果任何水平线与图像的交点多于一个，则该函数在其整个定义域上不可逆。 **Finding the inverse 求反函数的步骤**: 1. Write y = f(x) 2. Swap x and y: x = f(y) 3. Solve for y in terms of x 4. Write f⁻¹(x) = (your result) 5. State the domain of f⁻¹ (which equals the range of f)

📝 Important Property 重要性质

For any invertible function f:
f(f⁻¹(x)) = x for all x in the domain of f⁻¹
f⁻¹(f(x)) = x for all x in the domain of f
The domain of f⁻¹ = the range of f
The range of f⁻¹ = the domain of f

**The self-inverse function 自反函数**: Some functions equal their own inverse: f(x) = f⁻¹(x). Classic IB examples: f(x) = -x, f(x) = 1/x, f(x) = (ax+b)/(cx-a) for certain constants. These produce graphs symmetric about y = x. --- ### 5. Function Transformations 函数图像变换 Transformations are the single most heavily tested topic in IB Functions, appearing in both Paper 1 and Paper 2. 函数图像变换是IB考试中考查最频繁的知识点，卷一和卷二都会出现。

Translation 平移

f(x) + d
Vertical shift by d
垂直平移d个单位

f(x - c)
Horizontal shift by c (right)
水平平移c个单位(右)

Stretch 伸缩

a f(x)
Vertical stretch by factor a
垂直伸缩因子a

f(b x)
Horizontal stretch by 1/b
水平伸缩因子1/b

Reflection 反射

-f(x)
Reflect across x-axis
关于x轴对称

f(-x)
Reflect across y-axis
关于y轴对称

**The golden rule 黄金法则**: Apply transformations in this order 按照以下顺序应用变换: 1. Horizontal translation: f(x - c) 2. Horizontal stretch/reflection: f(b x) 3. Vertical stretch/reflection: a f(x) 4. Vertical translation: f(x) + d > ⚠️ **Common mistake 常见错误**: Students often mix up f(2x) and f(x/2). f(2x) COMPRESSES the graph horizontally (narrower); f(x/2) STRETCHES it (wider). f(2x)是水平压缩，f(x/2)是水平拉伸。 **Mapping method for transformations 变换的映射法**: Instead of memorising rules, use the mapping approach. For y = a f(b(x - c)) + d, map (x, y) to (x/b + c, a y + d). Every point on the original graph transforms according to this rule. This is especially powerful for HL questions where you need to find the image of a specific point after multiple transformations. --- ### 6. Key Function Types 核心函数类型 #### Linear Functions 线性函数: f(x) = mx + c The simplest yet fundamental. Slope m determines steepness; c is the y-intercept. 斜率m决定倾斜程度，c是y轴截距。 Two key forms: **gradient-intercept** (y = mx + c) and **point-slope** (y - y₁ = m(x - x₁)). For HL, you also need to work with **vector and parametric forms** of linear functions in the context of 3D geometry. #### Quadratic Functions 二次函数: f(x) = ax² + bx + c Three forms you MUST know 必须掌握的三种形式: - **Standard form** 一般式: ax² + bx + c -- shows y-intercept - **Vertex form** 顶点式: a(x - h)² + k -- shows vertex (h, k) directly - **Factored form** 因式分解式: a(x - p)(x - q) -- shows x-intercepts (roots) The **discriminant** Δ = b² - 4ac tells you the number of real roots: Δ > 0 (two roots), Δ = 0 (one root), Δ < 0 (no real roots). 判别式决定实根个数。 **Axis of symmetry**: x = -b/(2a) for standard form, or x = h for vertex form. This is the vertical line through the vertex. Every quadratic is symmetric about this axis. 对称轴经过顶点，二次函数关于对称轴对称。 #### Rational Functions 有理函数: f(x) = P(x)/Q(x) Key features: vertical asymptotes (where Q(x) = 0), horizontal asymptotes (compare degrees of P and Q), and holes or removable discontinuities (common factors). 核心特征：垂直渐近线、水平渐近线和可去间断点。 The **horizontal asymptote rule** 水平渐近线规则: - deg(P) < deg(Q): y = 0 - deg(P) = deg(Q): y = leading coefficient of P / leading coefficient of Q - deg(P) > deg(Q): no horizontal asymptote (check for oblique/slant asymptote) #### Exponential and Logarithmic Functions 指数与对数函数 f(x) = aˣ and f(x) = logₐ(x) are inverses. Key IB property: eˣ and ln(x) are the natural pair. 指数函数与对数函数互为反函数。The exponential has horizontal asymptote y = 0; the logarithm has vertical asymptote x = 0. **Laws you must know 必须掌握的对数法则**: - logₐ(xy) = logₐ(x) + logₐ(y) - logₐ(x/y) = logₐ(x) - logₐ(y) - logₐ(xⁿ) = n logₐ(x) - logₐ(1) = 0 and logₐ(a) = 1 - Change of base: logₐ(b) = log(b)/log(a) = ln(b)/ln(a) #### Trigonometric Functions 三角函数 sin(x), cos(x), tan(x) are periodic. For IB AA, you need to know exact values at key angles (0°, 30°, 45°, 60°, 90° and their radian equivalents), graph transformations of trig functions with amplitude and period changes, and the restricted domains that make arcsin, arccos, arctan functions possible. **The unit circle approach 单位圆方法**: Understand that sin(θ) is the y-coordinate and cos(θ) is the x-coordinate of a point on the unit circle. This visual understanding makes quadrant analysis and sign determination automatic rather than memorised. --- ### 7. The IB Exam: What They Actually Test 考试实际考查什么

Paper 1 (No Calculator) 卷一

Sketching functions by hand
Solving equations algebraically
Proving identities and properties
Exact values of trig functions
Finding inverses symbolically
Tip: Practice mental math and factoring daily. 每天练习心算和因式分解。

Paper 2 (GDC Allowed) 卷二

Solving complex equations using GDC
Finding intersections graphically
Regression and curve fitting
Transformations with parameters
Domain/range using GDC graphs
Tip: Master your GDC shortcuts. 熟练掌握计算器快捷键。

**IA connections IA内部评估关联**: Functions underpin many excellent IA topics. Consider investigating population growth models (exponential and logistic functions), projectile motion (quadratic functions with transformations), sound wave analysis (trigonometric functions with transformations), or cooling curves (exponential decay with translation). **Top 5 most tested topics 五大高频考点**: 1. Finding domain and range of rational or radical functions 求有理式和根式函数的定义域值域 2. Proving a function is injective and finding its inverse 证明单射性并求反函数 3. Describing a sequence of transformations 描述函数图像的变换序列 4. Composite function domain restrictions 复合函数的定义域限制 5. Solving f(x) = g(x) graphically and algebraically 图像法与代数法解方程 --- ### 8. Study Plan: 4 Week Sprint 四周冲刺计划

Week 1 第一周

Foundation 基础巩固
Function definition, domain, range
Linear and quadratic functions
Daily: 10 domain/range problems
Memorise exact trig values
Review: vertical/horizontal line tests

Week 2 第二周

Core Techniques 核心技法
Composite and inverse functions
Rational and radical functions
Daily: 5 composite + 5 inverse
Start timed Paper 1 sections
Master algebraic manipulation

Week 3 第三周

Applications 应用拓展
Exponential and log functions
Trigonometric functions in depth
Full transformation sequences
Daily: 10 transformation questions
Connect functions to real contexts

Week 4 第四周

Exam Readiness 考前冲刺
Full past papers under timed conditions
Mixed review: all function types
Error analysis on weak areas
Final: 2 full mock exams
Build exam stamina and pacing

**Daily routine 每日安排**: 45 minutes of focused practice, broken into 15 minutes of theory review, 20 minutes of problem solving, and 10 minutes of error analysis. Consistency beats cramming every time. 每天45分钟：15分钟复习理论，20分钟解题，10分钟分析错题。持续性远胜于临时抱佛脚。 --- ### 9. Calculator Guide 计算器使用指南 For IB Math AA, the Casio fx-9860GIII or TI-84 Plus CE are the standard GDC choices approved by the IB. For Chinese students who prefer a Chinese interface and need a reliable companion for daily practice and homework, the **Casio fx-991CN X** is an exceptional non-GDC calculator that handles function evaluation, equation solving, table generation, and matrix operations with ease. It supports 696 functions including complex numbers, statistical regression, and vector calculations -- all with a clear Chinese language display that reduces cognitive load during intense problem solving.

fx-991CN X 中文版计算器-老王推荐

¥168.00 🛒 微信小店购买 For function analysis specifically, here is how to leverage your calculator 如何使用计算器分析函数: - **TABLE mode**: Input f(x) and scan values to understand domain/range behaviour and spot asymptotes - **SOLVE mode**: Numerically solve f(x) = g(x) intersections when exact algebraic solutions are not required - **CALC function**: Evaluate f(a) at any specific x-value to verify your algebraic work - **GRAPH mode (GDC only)**: Visualise transformations by overlaying y = f(x) and y = a f(b(x - c)) + d --- ### 10. Common Mistakes to Avoid 避坑指南

❌ Forgetting domain restrictions: Always state domain for functions involving roots, logs, or denominators. Never assume "all real numbers." 忘记定义域限制：永远不要默认"全体实数"。

❌ Treating f⁻¹(x) as 1/f(x): f⁻¹ is the inverse function, NOT the reciprocal. These are completely different operations with different meanings and graphs. 把反函数符号当成倒数。

❌ Wrong transformation order: Horizontal changes come before vertical changes. Inside the function before outside. Mixing this up is the number one transformation error. 变换顺序错误：水平变换先于垂直变换。

❌ Forgetting the horizontal line test: A function must be one-to-one (injective) to be invertible on its entire domain. If it fails, you must restrict the domain. 忘记水平线检验就直接求反函数。

❌ Assuming f(g(x)) = g(f(x)): Composition is almost never commutative. Always verify with a counterexample rather than assuming equality. 错误假设复合函数可交换。

❌ Confusing "x-intercept" with "y-intercept": x-intercept is where y = 0 (set f(x) = 0). y-intercept is where x = 0 (evaluate f(0)). 混淆x轴截距和y轴截距。

--- ### 11. Practice Questions 自测练习 Try these before your exam. Answers are provided below each question. 考试前试试这些题目，每题后面附有答案和思路。 **Q1**: Find the domain of f(x) = ln(x² - 4) / √(x + 3). State your answer in interval notation. 求定义域，用区间表示法写出答案。 > A1: Need x² - 4 > 0 so x < -2 or x > 2; and x + 3 > 0 so x > -3. Intersection: x ∈ (-3, -2) ∪ (2, ∞). **Q2**: Let f(x) = 2x + 1 and g(x) = 3/(x - 2). Find (f ∘ g)(x) and state its domain. > A2: (f ∘ g)(x) = 2[3/(x - 2)] + 1 = 6/(x - 2) + 1. g domain: x ≠ 2; g(x) must be in f domain (all reals). Domain: x ≠ 2. **Q3**: Prove that f(x) = (x + 2)/(x - 2), x > 2 is injective. Find f⁻¹ and verify f(f⁻¹(x)) = x. > A3: If f(a) = f(b), then (a+2)/(a-2) = (b+2)/(b-2). Cross multiply: (a+2)(b-2) = (b+2)(a-2). Expand: ab - 2a + 2b - 4 = ab - 2b + 2a - 4. Cancel: -2a + 2b = 2a - 2b so 4b = 4a so a = b. Injectivity proven. For inverse: y = (x+2)/(x-2), swap: x = (y+2)/(y-2), solve: x(y-2) = y+2, xy - 2x = y + 2, xy - y = 2x + 2, y(x-1) = 2x + 2, y = 2(x+1)/(x-1). So f⁻¹(x) = 2(x+1)/(x-1). Verify: f(f⁻¹(x)) = [2(x+1)/(x-1) + 2] / [2(x+1)/(x-1) - 2] = ... = x. ✓ **Q4**: Describe the transformations that map y = x² onto y = 3(x + 1)² - 4. > A4: Start from y = x². (x + 1)²: horizontal translation 1 unit LEFT. 3(x + 1)²: vertical stretch by factor 3. 3(x + 1)² - 4: vertical translation 4 units DOWN. **Q5**: The graph of y = f(x) has vertex at (2, 3). Where is the vertex of y = 2f(3x - 6) + 1? > A5: Rewrite as y = 2f(3(x - 2)) + 1. Original (2, 3). x: divided by 3 (horizontal stretch factor 1/3), then +2: 2/3 + 2 = 8/3. y: multiplied by 2, then +1: 3*2+1 = 7. New vertex: (8/3, 7). --- *Mastering functions requires consistent practice and conceptual clarity. 掌握函数需要持续的练习和清晰的概念理解。 Every hour you invest in truly understanding why transformations work the way they do pays back tenfold in the exam hall. Functions are not a topic to memorise; they are a language to internalise. 函数不是需要死记硬背的知识点，而是一种需要内化的数学语言。* *For those using the fx-991CN X: use TABLE mode to explore domain and range behaviour numerically, SOLVE mode for tricky equations, and the CALC function to verify your algebraic evaluations. The Chinese interface makes it especially accessible for bilingual students who want to minimise friction during intense problem solving. 中文界面在双语学习环境中显著降低了操作门槛，让你把精力集中在数学思维上。* --- 🎓 **Need personalised IB Math AA tutoring? 需要IB数学AA个性化辅导？** Professional bilingual instruction tailored to SL and HL syllabi. 专业中英双语教学，精准对标SL和HL考纲。 📲 **WeChat: 16621398022** One-on-one and small group sessions available. 一对一及小班课均可安排。

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