4 Functions | 函数 - aleveler.com

📚 4 Functions | 函数

Functions form the backbone of the IB Mathematics curriculum, bridging algebra, geometry, and real-world modelling. A solid grasp of functions—their notation, transformations, inverses, and key families—is essential for success in both Standard Level and Higher Level. This article unpacks Topic 4 from the IB syllabus, presenting core concepts and techniques that will sharpen your problem-solving skills and prepare you for examination-style questions.

函数是 IB 数学课程的骨干，连接代数、几何和现实世界建模。牢固掌握函数的记号、变换、反函数和重要函数族，对于在标准水平和高级水平中取得成功至关重要。本文将拆解 IB 大纲中的主题 4，介绍核心概念和技巧，助你提升解题能力，为考试题型做好准备。

1. Definition of a Function | 函数的定义

A function is a special relation where each input from the domain is assigned exactly one output in the range. If an input maps to two different outputs, the relation is not a function. In IB, we often test this with vertical lines on a graph.

函数是一种特殊的关系，其中定义域中的每个输入恰好对应值域中的一个输出。如果一个输入对应两个不同的输出，则该关系不是函数。在 IB 中，我们通常用图像上的竖直线来检验。

The vertical line test states that if any vertical line intersects a graph more than once, the graph does not represent a function. For example, a circle fails the test, while a parabola passes it.

竖直线检验指出，如果任何一条竖直线与图像相交超过一次，则该图像不表示函数。例如，圆的图像不通过检验，而抛物线则通过。

Functions can be ‘one-to-one’ (each output comes from exactly one input) or ‘many-to-one’ (several inputs give the same output). The IB syllabus requires you to distinguish these types by looking at mappings or graphs.

函数可以是”一一对应”（每个输出来自恰好一个输入）或”多对一”（多个输入给出相同输出）。IB 大纲要求你通过映射或图像区分这些类型。

2. Function Notation, Domain and Range | 函数记号与定义域和值域

We write f(x) to denote the output of a function f for an input x. The domain is the set of all possible input values, while the range is the set of all output values actually produced. IB problems frequently ask for the maximal domain of a function given its formula.

我们用 f(x) 表示函数 f 对输入 x 的输出。定义域是所有可能输入值的集合，而值域是所有实际产生的输出值的集合。IB 题目经常要求根据表达式求函数的最大定义域。

To find the maximal domain of f(x) = √(x – 3), for instance, we require the radicand x – 3 ≥ 0, so the domain is x ≥ 3. For rational functions like f(x) = 1/(x – 2), the denominator cannot be zero, so x ≠ 2.

例如，求 f(x) = √(x – 3) 的最大定义域，要求被开方数 x – 3 ≥ 0，因此定义域为 x ≥ 3。对于有理函数如 f(x) = 1/(x – 2)，分母不能为零，所以 x ≠ 2。

Range determination often relies on graph analysis or completing the square for quadratics. Writing f(x) = (x – 1)² + 2 immediately reveals that the minimum output is 2, so the range is y ≥ 2.

值域的确定通常依赖于图像分析或对二次函数进行配平方。将 f(x) = (x – 1)² + 2 写成顶点式，立刻可知最小输出为 2，因此值域为 y ≥ 2。

3. Composite Functions | 复合函数

Given two functions f and g, the composite function (f ∘ g)(x) means we first apply g to x, then apply f to the result. Symbolically, (f ∘ g)(x) = f(g(x)). The order is crucial: f(g(x)) is generally not the same as g(f(x)).

给定两个函数 f 和 g，复合函数 (f ∘ g)(x) 表示先将 g 作用于 x，再将 f 作用于所得结果。符号上 (f ∘ g)(x) = f(g(x))。顺序至关重要：f(g(x)) 通常不同于 g(f(x))。

IB exam questions commonly ask you to write an expression for f(g(x)), simplify it, and then state its domain. The domain of the composite function consists of all x such that x is in the domain of g and g(x) is in the domain of f.

IB 考题通常会要求你写出 f(g(x)) 的表达式、化简，然后说出其定义域。复合函数的定义域包含所有满足 x 在 g 的定义域内且 g(x) 在 f 的定义域内的 x。

For example, if f(x) = ln x and g(x) = 2x – 5, then f(g(x)) = ln(2x – 5). The domain requires 2x – 5 > 0, leading to x > 2.5. Paying attention to restrictions avoids losing marks.

例如，若 f(x) = ln x 且 g(x) = 2x – 5，则 f(g(x)) = ln(2x – 5)。定义域要求 2x – 5 > 0，得到 x > 2.5。留意限制条件可避免失分。

4. Inverse Functions | 反函数

An inverse function f⁻¹ reverses the effect of f. It exists only if f is one-to-one; we often use the horizontal line test on the graph of f to check this. If any horizontal line cuts the graph more than once, the function is not invertible over its entire domain.

反函数 f⁻¹ 将 f 的效果逆转。只有当 f 是一一对应时反函数才存在；我们常在 f 的图像上用水平线检验来判断。若任何水平线切割图像超过一次，则该函数在其整个定义域上不可逆。

To find an inverse algebraically, write y = f(x), swap x and y, then solve for y. The resulting expression gives f⁻¹(x). The graphs of f and f⁻¹ are symmetric with respect to the line y = x.

要代数求出反函数，先写出 y = f(x)，交换 x 和 y，然后解出 y。得到的表达式即为 f⁻¹(x)。f 和 f⁻¹ 的图像关于直线 y = x 对称。

IB questions sometimes restrict the domain of a non‑invertible function (e.g., y = x² restricted to x ≥ 0) so that its inverse exists. You must be comfortable stating the domain and range of the inverse as well.

IB 题目有时会限制不可逆函数的定义域（如 y = x² 限制为 x ≥ 0）以使反函数存在。你需要熟练说出反函数的定义域和值域。

5. Transformations of Graphs | 图像的变换

Transformations allow us to sketch complicated functions by modifying basic graphs. The main types are translations, stretches, and reflections. For IB, you must know the precise effect of each parameter on the graph of y = f(x).

变换使我们能通过修改基本图像来绘制复杂函数。主要类型有平移、伸缩和反射。对 IB 而言，你必须知道每个参数对 y = f(x) 图像的确切影响。

Vertical translation: y = f(x) + k shifts the graph up (k > 0) or down (k < 0). Horizontal translation: y = f(x - h) shifts it right (h > 0) or left (h < 0).

竖直平移：y = f(x) + k 将图像向上 (k > 0) 或向下 (k < 0) 移动。水平平移：y = f(x - h) 将其向右 (h > 0) 或向左 (h < 0) 移动。

Vertical stretch: y = a f(x) with a > 1 stretches vertically, while 0 < a < 1 compresses. Horizontal stretch: y = f(bx) with b > 1 compresses horizontally, while 0 < b < 1 stretches.

竖直伸缩：y = a f(x) 中 a > 1 为竖直拉伸，0 < a < 1 为竖直压缩。水平伸缩：y = f(bx) 中 b > 1 为水平压缩，0 < b < 1 为水平拉伸。

Reflections: y = -f(x) reflects in the x-axis; y = f(-x) reflects in the y-axis. Being systematic with the order of transformations in IB questions is essential—usually stretches first, then translations.

反射：y = -f(x) 为关于 x 轴反射；y = f(-x) 为关于 y 轴反射。在 IB 题目中，系统地进行变换顺序至关重要——通常先做伸缩，再做平移。

6. Quadratic Functions | 二次函数

Quadratic functions are of the form f(x) = ax² + bx + c, where a ≠ 0. Their graphs are parabolas. IB expects you to move fluently between standard form, vertex form f(x) = a(x – h)² + k, and factorised form f(x) = a(x – p)(x – q).

二次函数形如 f(x) = ax² + bx + c，其中 a ≠ 0。其图像为抛物线。IB 希望你熟练地在标准式、顶点式 f(x) = a(x – h)² + k 和分解式 f(x) = a(x – p)(x – q) 之间转换。

The discriminant Δ = b² – 4ac determines the number of real roots: two real roots if Δ > 0, one repeated root if Δ = 0, and no real roots if Δ < 0. This is frequently tested alongside graph sketching.

判别式 Δ = b² – 4ac 决定实数根的个数：Δ > 0 时有两个实根，Δ = 0 时有一个重根，Δ < 0 时无实根。这一知识点常与图像绘制结合考查。

The axis of symmetry is x = -b/(2a), and the vertex is at this x-value. Completing the square reveals the vertex (h, k) and makes solving equations and finding ranges straightforward.

对称轴为 x = -b/(2a)，顶点即在该 x 值处。配平方可揭示顶点 (h, k)，并使解方程和求值域变得直接明了。

7. Polynomial Functions | 多项式函数

Polynomial functions generalise quadratics to higher degrees. IB Mathematics includes factorisation, the Remainder and Factor Theorems, and analysis of end behaviour. A polynomial of degree n has at most n turning points and at most n real roots.

多项式函数将二次函数推广到更高次。IB 数学涵盖因式分解、余数定理和因式定理，以及末端行为分析。n 次多项式最多有 n 个驻点，最多有 n 个实根。

The Factor Theorem states that (x – p) is a factor of P(x) if and only if P(p) = 0. This is a powerful tool for solving higher-degree equations and factorising cubics and quartics.

因式定理指出，(x – p) 是 P(x) 的因式当且仅当 P(p) = 0。这是解高次方程和分解三次及四次因式的有力工具。

For large |x|, the term with the highest power dominates. If the leading coefficient is positive and the degree is even, both ends of the graph go to +∞. If odd and positive, the left goes to -∞ and the right to +∞.

当 |x| 很大时，最高次项占主导。若首项系数为正且次数为偶，图像两端都趋向 +∞。若次数为奇且首项系数为正，左端趋向 -∞，右端趋向 +∞。

8. Rational Functions | 有理函数

Rational functions have the form f(x) = P(x)/Q(x), where P and Q are polynomials. Asymptotes are central to sketching these functions. Vertical asymptotes occur where Q(x) = 0 and the function is undefined.

有理函数具有 f(x) = P(x)/Q(x) 的形式，其中 P 和 Q 是多项式。渐近线是绘制这些函数图像的核心。垂直渐近线出现在 Q(x) = 0 且函数未定义之处。

Horizontal asymptotes describe end behaviour. If the degree of P is less than that of Q, the horizontal asymptote is y = 0. If degrees are equal, y equals the ratio of leading coefficients. If the degree of P exceeds that of Q by one, a slant asymptote appears.

水平渐近线描述末端行为。若 P 的次数小于 Q 的次数，水平渐近线为 y = 0。若次数相等，y 等于首项系数之比。若 P 的次数比 Q 的次数大 1，则出现斜渐近线。

IB questions expect you to find all asymptotes, intercepts, and holes (removable discontinuities) in order to sketch the graph. Simplifying a rational expression can reveal a hole at a value where both numerator and denominator become zero.

IB 题目要求你找出所有渐近线、截距和可去间断点（洞）以便绘制图像。化简有理式可揭示分子分母同时为零处的洞。

9. Exponential and Logarithmic Functions | 指数与对数函数

Exponential functions f(x) = aˣ (a > 0, a ≠ 1) model growth and decay. Their graphs pass through (0, 1), have a horizontal asymptote y = 0, and are always increasing for a > 1 or decreasing for 0 < a < 1. The base e ≈ 2.718 is widely used in IB problems.

指数函数 f(x) = aˣ (a > 0, a ≠ 1) 用于建模增长和衰减。其图像经过 (0, 1)，有水平渐近线 y = 0，当 a > 1 时始终递增，0 < a < 1 时始终递减。底数 e ≈ 2.718 在 IB 题目中被广泛使用。

Logarithmic functions f(x) = logₐ x are the inverses of exponentials. Their domain is x > 0, vertical asymptote x = 0, and they pass through (1, 0). The natural logarithm ln x means logₑ x. The laws of logs are essential for solving equations.

对数函数 f(x) = logₐ x 是指数函数的反函数。其定义域为 x > 0，垂直渐近线为 x = 0，且经过点 (1, 0)。自然对数 ln x 即 logₑ x。对数律是解方程的关键。

Because exponentials and logs are inverses, e^(ln x) = x and ln(eˣ) = x. IB questions regularly combine these with algebra to solve equations like 2ˣ = 7, giving x = log₂ 7, or e²ˣ⁻¹ = 5.

由于指数函数和对数函数互为反函数，有 e^(ln x) = x 和 ln(eˣ) = x。IB 题目通常将这些与代数相结合来解诸如 2ˣ = 7 的方程，得到 x = log₂ 7，或解 e²ˣ⁻¹ = 5。

10. Trigonometric Functions | 三角函数

The three primary trigonometric functions in IB are sine, cosine, and tangent. Their graphs repeat periodically: sin x and cos x have period 2π, amplitude 1; tan x has period π and vertical asymptotes at odd multiples of π/2.

IB 中的三个主要三角函数为正弦、余弦和正切。它们的图像呈周期性重复：sin x 和 cos x 的周期为 2π，振幅为 1；tan x 的周期为 π，并在 π/2 的奇数倍处有垂直渐近线。

General sinusoidal models take the form f(x) = A sin(B(x – C)) + D. Here, |A| is amplitude, 2π/|B| is period, C is horizontal shift, and D is vertical shift. You must be able to deduce these parameters from a graph.

一般的正弦型模型为 f(x) = A sin(B(x – C)) + D。其中 |A| 为振幅，2π/|B| 为周期，C 为水平位移，D 为竖直位移。你必须能从图像中推导这些参数。

Exact values for special angles (0, π/6, π/4, π/3, π/2 and their multiples) must be memorised. Trigonometric identities like sin²θ + cos²θ = 1 are heavily tested, as are double-angle formulas for HL students.

特殊角（0, π/6, π/4, π/3, π/2 及其倍数）的精确值必须记住。三角恒等式如 sin²θ + cos²θ = 1 常被考查，HL 学生还要掌握倍角公式。

11. Solving Equations Graphically | 图像法解方程

IB strongly emphasises using your graphical display calculator (GDC) to solve equations. You can find the roots of f(x) = 0 by locating the x-intercepts of y = f(x). For equations like f(x) = g(x), you graph both sides and find intersection points.

IB 强烈强调使用图像显示计算器 (GDC) 解方程。你可通过确定 y = f(x) 的 x 截距来找出 f(x) = 0 的根。对于 f(x) = g(x) 这类方程，你可画出两边的图像并找出交点。

When solving equations without GDC, transformations and known function properties are key. For example, √(x + 3) = x – 1 can be solved algebraically by squaring, but you must check for extraneous solutions introduced by squaring.

在不使用 GDC 的情况下解方程时，变换和已知函数性质是关键。例如，√(x + 3) = x – 1 可通过平方代数求解，但必须检查因平方引入的增根。

Graphical approaches also help with inequalities. f(x) > k means you identify where the graph of f lies above the line y = k. This method gives clear interval notation solutions.

图像法也有助于解不等式。f(x) > k 意味着识别 f 的图像位于直线 y = k 之上的部分。这种方法能给出清晰的区间表示解。

12. Applications of Functions | 函数的应用

Functions model real-life phenomena such as population growth, radioactive decay, projectile motion, and economic cost. IB Paper 2 and the IA exploration both require you to select, fit, and critique an appropriate function model.

函数可建模现实现象，如人口增长、放射性衰变、抛体运动和经济成本。IB 卷二和内部评估 (IA) 探究都要求你选择、拟合和评价合适的函数模型。

A typical application: the height of a ball thrown upwards can be modelled by a quadratic h(t) = -4.9t² + vt + h₀. Finding the maximum height means locating the vertex; finding the time it hits the ground means solving h(t) = 0.

一个典型应用：向上抛出的球的高度可用二次函数 h(t) = -4.9t² + vt + h₀ 建模。求最大高度即找出顶点；求落地时间即解 h(t) = 0。

Exponential models A₀eᵏᵗ appear in finance (continuous compounding) and half-life problems. Logarithmic models often describe the decibel scale or pH. In IB, you are expected to interpret parameters and constants in context.

指数模型 A₀eᵏᵗ 出现在金融（连续复利）和半衰期问题中。对数模型常用于描述分贝量表或 pH 值。在 IB 中，你需要结合情境解释参数和常量。

Optimisation problems use functions to maximise area or minimise cost, often by expressing one variable in terms of another and then applying differentiation (HL) or graphical methods. Even at SL, you can find maxima via the GDC trace feature.

优化问题利用函数求最大面积或最小成本，通常通过将一个变量用另一个变量表示，然后应用微分 (HL) 或图像法来解决。即使是在 SL 水平，你也可以通过 GDC 的 trace 功能找到最大值。

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4 Functions | 函数