📚 IB Mathematics: Essential Functions & Problem-Solving Techniques | IB数学：基本函数与解题技巧全解析
Functions are the backbone of IB Mathematics, appearing in every paper and connecting algebra, geometry, and calculus. This guide unpacks the essential types of functions you must master, along with proven problem-solving strategies for both Analysis & Approaches (AA) and Applications & Interpretation (AI).
函数是IB数学的支柱，出现在每一份试卷中，并将代数、几何与微积分紧密相连。本指南将拆解你必须掌握的基本函数类型，并为你提供适用于数学分析与方法（AA）以及数学应用与解释（AI）的解题策略。
1. Function Basics: Domain, Range, and Notation | 函数基础：定义域、值域与记号
A function is a rule that assigns exactly one output to each input from its domain. The domain is the set of all possible input values, while the range is the set of all resulting output values.
函数是一种规则，它将定义域中的每一个输入值唯一对应到一个输出值。定义域是所有可能输入值的集合，而值域是所有输出结果构成的集合。
Function notation 'f(x)' emphasises the dependence on 'x'. For example, f(x) = 2x + 3 means 'double the input, then add 3'.
Function notation：“f(x)”强调了对“x”的依赖。例如，f(x) = 2x + 3 表示“将输入乘以2，再加3”。
Natural domain: exclude values that make denominators zero, radicands of even roots negative, or arguments of logarithms non-positive.
自然定义域：排除使分母为零、偶次根号内为负数或对数真数非正的值。
Domain of f(x) = √x − 2: x ≥ 2
When finding the range, always consider the graph's shape, its turning points, and any horizontal or vertical asymptotes.
求值域时，始终要结合图像的形状、极值点以及水平或垂直渐近线来判断。
2. Linear Functions and Lines | 线性函数与直线
Linear functions take the form f(x) = mx + c, where 'm' is the slope and 'c' is the y-intercept. They grow at a constant rate and have a straight-line graph.
线性函数的形式为 f(x) = mx + c，其中“m”是斜率，“c”是纵截距。它们以恒定速率增长，图像是一条直线。
Key formulas include the gradient between two points and the equation of a line:
关键公式包括两点间的斜率公式以及直线的方程：
m = (y₂ − y₁) ⁄ (x₂ − x₁)
y − y₁ = m(x − x₁)
For perpendicular lines, the product of their gradients is −1; for parallel lines, the gradients are equal.
两直线垂直时，斜率之积为 −1；两直线平行时，斜率相等。
3. Quadratic Functions | 二次函数
with a ≠ 0. Their graphs are parabolas, which open upward if a > 0 and downward if a < 0.
二次函数的一般形式为 f(x) = ax² + bx + c，其中 a ≠ 0。其图像是抛物线：当 a > 0 时开口向上，当 a < 0 时开口向下。
The three common forms each reveal different features:
三种常见形式各自揭示不同的特征：
General form (x) = ax² + bx + c: useful for finding the y-intercept and applying the quadratic formula.
一般形式 f(x) = ax² + bx + c：便于求纵截距以及套用求根公式。
Vertex form f(x) = a(x − h)² + k: directly gives the vertex (h, k).
顶点式 f(x) = a(x − h)² + k：直接给出顶点 (h, k)。
Factored form f(x) = a(x − p)(x − q): reveals the x-intercepts p and q.
交点式 f(x) = a(x − p)(x − q)：直接给出与x轴的交点 p 和 q。
The quadratic formula is essential for solving equations when factorization is difficult:
当因式分解困难时，求根公式是解方程的关键工具：
x = (−b ± √b² − 4ac) ⁄ (2a)
0 gives two distinct roots, Δ = 0 gives one repeated root, and Δ < 0 gives no real roots.
判别式 Δ = b² − 4ac 决定实数根的个数：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 没有实数根。
4. Rational Functions and Asymptotes | 有理函数与渐近线
x − 3). Their graphs often feature asymptotes that the curve approaches but never touches.
有理函数是两个多项式的商，例如 f(x) = (2x + 1) ⁄ (x − 3)。其图像通常具有曲线无限接近却不会触及的渐近线。
Vertical asymptote re occurs where the denominator equals zero and the numerator is non-zero: x = 3 for the example above.
垂直渐近线出现在分母为零且分子不为零的位置：上例中为 x = 3。
Horizontal asymptote depends on the degrees of numerator and denominator. If degrees are equal, the asymptote is the ratio of leading coefficients.
水平渐近线取决于分子与分母的次数。若次数相同，渐近线为最高次项系数之比。
For f(x) = (2x + 1) ⁄ (x − 3), horizontal asymptote: y = 2
When solving inequalities involving rational functions, always consider the sign changes around critical points and exclude values where the denominator is zero.
求解涉及有理函数的不等式时，必须考虑关键点周围的符号变化，并排除分母为零的值。
5. Exponential and Logarithmic Functions | 指数函数与对数函数
Exponential functions have the form f(x) = a · bˣ, where b > 0 and b ≠ 1. They model growth or decay with a constant percentage rate.
指数函数的形式为 f(x) = a · bˣ，其中 b > 0 且 b ≠ 1。它们以恒定的百分比速率描述增长或衰减。
Logarithmic functions are their inverses: if y = bˣ then x = logₐ(y). The key logarithm rules are essential for solving exponential equations:
对数函数是指数函数的反函数：若 y = bˣ，则 x = logₐ(y)。解决指数方程离不开以下对数运算法则：
logₐ(mn) = logₐ m + logₐ n
logₐ(m ⁄ n) = logₐ m − logₐ n
logₐ(mⁿ) = n logₐ m
, set up the model f(t) = A₀ · (1 + r)ᵗ and solve using natural logarithms when necessary.
对于IB中涉及半衰期或复利的问题，先建立模型 f(t) = A₀ · (1 + r)ᵗ，必要时使用自然对数求解。
6. Trigonometric Functions | 三角函数
Trigonometric functions such as sine, cosine, and tangent are periodic, making them ideal for modelling waves and oscillations.
正弦、余弦和正切等三角函数具有周期性，因此非常适合模拟波浪和振动。
The general form f(x) = A sin(B(x − C)) + D has key parameters:
一般形式 f(x) = A sin(B(x − C)) + D 包含以下关键参数：
A is the amplitude: half the vertical distance between maximum and minimum.
A 是振幅：最大值与最小值之间垂直距离的一半。
B affects the period: Period = 2π ⁄ B.
B 影响周期：周期 = 2π ⁄ B。
C is the horizontal shift (phase shift).
C 是水平平移（相位移动）。
D is the vertical shift, moving the midline.
D 是垂直平移，移动中线。
d identities, such as sin²θ + cos²θ = 1, are expected knowledge in both AA and AI courses.
常见的特殊值以及恒等式，如 sin²θ + cos²θ = 1，是AA和AI课程中要求掌握的内容。
7. Transformations of Functions | 函数的变换
Transformations allow you to sketch complicated functions by modifying a known base graph. The general rules are:
函数变换让你通过修改已知的基本图像来绘制复杂函数。一般规则如下：
y = f(x) + a → vertical translation by a
y = f(x − a) → horizontal translation by a
y = −f(x) → reflection in the x-axis
y = f(−x) → reflection in the y-axis
lways apply transformations to the correct variable: replace x or y in the original equation.
垂直伸缩 k 倍表示为 y = k f(x)，而水平伸缩 1 ⁄ k 倍表示为 y = f(kx)。始终对正确的变量施加变换：在原始方程中替换 x 或 y。
8. Composite Functions and Inverse Functions | 复合函数与反函数
Composite functions apply one function to the output of another. The notation (f ∘ g)(x) = f(g(x)) means 'apply g first, then f'.
复合函数将后一个函数作用于前一个函数的输出。记号 (f ∘ g)(x) = f(g(x)) 表示“先施加 g，再施加 f”。
To find the inverse f⁻¹(x), swap x and y, solve for y, and ensure the result is a function. A function has an inverse only if it is one-to-one; otherwise, restrict its domain.
求反函数 f⁻¹(x) 的步骤是：交换 x 和 y，解出 y，并确保结果仍是一个函数。函数只有双射时才存在反函数；否则需要限制定义域。
f(x) = 2x + 1 → f⁻¹(x) = (x − 1) ⁄ 2
The inverse of an exponential is a logarithm, and vice versa. This symmetry is heavily tested in IB papers.
指数函数的反函数是对数函数，反之亦然。这一对称关系在IB试卷中频繁考查。
9. Solving Equations Graphically and Analytically | 图像法与解析法求解方程
Some equations cannot be solved by simple algebra, so IB students must be comfortable with graphical solutions using a GDC (graphical display calculator).
某些方程无法通过简单代数求解，因此IB学生必须熟练使用图形计算器（GDC）进行图像求解。
For an equation like eˣ d find their intersection points. Alternatively, rearrange to f(x) = 0 and look for zeros.
对于像 eˣ = 5 − x² 这样的方程，你可以绘制两个函数的图像并找到交点；或者将方程整理为 f(x) = 0 并求零点。
Analytical approach: isolate the unknown, take logarithms, factor, or use the quadratic formula.
解析法：分离未知数、取对数、因式分解或使用求根公式。
Graphical approach: use GDC to find intersections, maximum/minimum points, or areas under curves.
图像法：使用GDC求交点、最大值/最小值点或曲线下方的面积。
Always check whether a solution lies within the given domain, especially for trigonometric or logarithmic equations.
务必检查解是否位于给定定义域内，尤其对于三角方程或对数方程。
10. Common Problem-Solving Techniques | 通用解题技巧
Effective problem-solving in IB functions requires a strategic mindset. Start by identifying the function type and the information you have: graph, equation, table, or verbal description.
在IB函数问题中高效解题需要策略性思维。首先判断函数类型以及你掌握的信息：图像、方程、表格还是文字描述。
Step 1: Identify the type of function.
Step 2: Extract key features (intercepts, vertex, asymptotes).
Step 3: Set up equations using given conditions.
Step 4: Solve and verify within domain/range.
r example, if a problem says a population doubles every 3 hours, model it as P(t) = P₀ · 2t⁄3.
对于应用题，先明确定义变量并建立模型。例如，若题目说人口每3小时翻一番，则建立模型 P(t) = P₀ · 2t⁄3。
11. Exam Tips and Common Pitfalls | 考试技巧与常见误区
Many students lose marks not because they cannot solve the problem, but because they skip necessary steps or forget domain restrictions.
许多学生丢分并非因为不会解题，而是因为跳过必要步骤或忘记定义域限制。
| Common Mistake | 常见误区 | Correction | 正确做法 |
| Forgetting that the denominator cannot be zero. | Always exclude values that make the denominator zero. |
| 忽略分母不能为零。 | 始终排除使分母为零的值。 |
| Taking the square root of both sides without ±. | Write x = ±√k when solving x² = k. |
| 对两边开平方时遗漏 ±。 | 解 x² = k 时写出 x = ±√k。 |
| Confusing horizontal and vertical transformations. | Remember: f(x − c) shifts right, f(x + c) shifts left. |
| 混淆水平与垂直变换。 | 记住：f(x − c) 右移，f(x + c) 左移。 |
In calculator papers, write down the equation you enter in the GDC. In non-calculator papers, show all algebraic steps clearly to earn method marks.
在可使用计算器的试卷中，写下输入GDC的方程；在不使用计算器的试卷中，清晰展示所有代数步骤以获取方法分。
12. Final Thoughts: Master the Basics, Then Build Up | 总结：掌握基础，层层递进
Functions connect every area of IB mathematics. By mastering domain and range, key function families, transformations, and inverse operations, you build a solid foundation for calculus, statistics, and modelling questions.
函数连接着IB数学的每一个领域。通过掌握定义域与值域、基本函数族、变换和反函数，你将为进一步学习微积分、统计与建模问题奠定坚实基础。
Regular practice with past papers is the single most effective way to internalise these techniques. For every question you practise, ask yourself: What type of function is this? What features matter? Which method is fastest?
定期练习历年真题是内化这些技巧的最有效途径。对每一道练习过的题目，都问自己：这是什么类型的函数？哪些特征最关键？哪种方法最快？
With consistent effort and a clear problem-solving framework, you can confidently approach any function question the IB exam throws at you.
只要坚持不懈地努力，并掌握清晰的解题框架，你就能自信地应对IB考试中任何函数问题。
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