Mathematics for the IB Diploma: Mastering Functions and Differentiation | IB文凭数学：掌握函数与微分

📚 Mathematics for the IB Diploma: Mastering Functions and Differentiation | IB文凭数学：掌握函数与微分

Functions and differentiation form the backbone of the IB Mathematics: Analysis and Approaches syllabus. Whether you are preparing for Standard Level or Higher Level, a strong command of function behaviour, domain and range, inverse and composite functions, and the rules of differentiation is essential. This revision guide walks through the core concepts, common problem types, and exam-ready strategies so you can approach questions with confidence.

函数与微分是 IB 数学：分析与方法（Analysis and Approaches）课程的核心支柱。无论你准备标准级别（SL）还是高级别（HL），扎实掌握函数的性态、定义域与值域、反函数与复合函数以及微分法则都至关重要。本复习指南将梳理核心概念、常见题型和应试策略，帮助你自信应对考试。

1. Understanding Functions, Domain and Range | 理解函数、定义域与值域

A function f maps each input x in its domain to exactly one output f(x). The domain is the set of allowed input values, while the range is the set of possible output values. For example, for f(x) = √(x − 2), the expression under the square root must be non-negative, so the domain is x ≥ 2. The range is y ≥ 0 because a square root never gives a negative output. To find a range, sketch the graph or analyse how the function transforms basic parent functions.

函数 f 将其定义域中的每个输入 x 唯一映射到一个输出 f(x)。定义域是允许输入值的集合，值域是可能输出值的集合。例如，对于 f(x) = √(x − 2)，平方根内的表达式必须非负，因此定义域为 x ≥ 2。值域为 y ≥ 0，因为平方根永远不会产生负输出。求值域时，可以通过画图或分析函数如何由基本母函数变换得到。

When identifying the domain, look for three main restrictions: denominators cannot be zero, radicands of even roots must be non-negative, and logarithms require positive arguments. For quadratics, the range can be found efficiently by completing the square, since the vertex gives the maximum or minimum output value.

确定定义域时，主要检查三类限制：分母不能为零；偶次根式的被开方数必须非负；对数只接受正数。对于二次函数，可以通过配方法高效地求出值域，因为顶点给出了输出的最大值或最小值。

• Domain restrictions: denominators ≠ 0, radicands ≥ 0, logarithms of positive numbers only. 定义域限制：分母 ≠ 0，被开方数 ≥ 0，对数只取正数。
• Range can be found using completing the square for quadratics. 二次函数值域可通过配方法求得。

2. Function Notation and Transformations | 函数记号与图像变换

Function notation f(x) allows us to describe transformations efficiently. For a base function f(x), the general transformed function is g(x) = a f(b(x − c)) + d. The parameter a gives a vertical stretch by factor |a| and reflects in the x-axis if a < 0. The parameter b gives a horizontal stretch by factor 1/|b| and reflects in the y-axis if b < 0. The parameter c shifts the graph horizontally by c units, and d shifts it vertically by d units.

函数记号 f(x) 使我们能够高效描述图像变换。对于基本函数 f(x)，一般变换后的函数为 g(x) = a f(b(x − c)) + d。参数 a 表示以 |a| 为倍数的垂直伸缩，若 a < 0 则关于 x 轴反射。参数 b 表示以 1/|b| 为倍数的水平伸缩，若 b < 0 则关于 y 轴反射。参数 c 使图像水平平移 c 个单位，d 使图像垂直平移 d 个单位。

g(x) = a f(b(x − c)) + d

For example, starting with f(x) = x², the transformed function h(x) = −2(x + 3)² + 1 involves a reflection in the x-axis, a vertical stretch by factor 2, a horizontal shift 3 units left, and a vertical shift 1 unit up. The vertex therefore moves from (0, 0) to (−3, 1).

例如，以 f(x) = x² 为出发点，变换后的函数 h(x) = −2(x + 3)² + 1 包含关于 x 轴反射、垂直伸长为原来的 2 倍、水平向左平移 3 个单位以及垂直向上平移 1 个单位。因此顶点从 (0, 0) 移动到 (−3, 1)。

• Horizontal shifts move the graph left/right and are caused by changing x to x − c. 水平平移由将 x 变为 x − c 引起，图像左右移动。
• Vertical shifts move the graph up/down and are caused by adding d outside the function. 垂直平移由在函数外部加上 d 引起，图像上下移动。

3. Composite and Inverse Functions | 复合函数与反函数

A composite function (f ∘ g)(x) = f(g(x)) applies g first, then f. It is only defined when the output of g lies in the domain of f. An inverse function f⁻¹(x) reverses the action of f; it exists only if f is one-to-one on its domain. To find an inverse, write y = f(x), swap x and y, then solve for y. For f(x) = 2x + 3, we have y = 2x + 3, swap to x = 2y + 3, so y = (x − 3)/2, and f⁻¹(x) = (x − 3)/2.

复合函数 (f ∘ g)(x) = f(g(x)) 先应用 g，再应用 f。只有当 g 的输出落在 f 的定义域内时，复合函数才有定义。反函数 f⁻¹(x) 反转 f 的作用；只有当 f 在其定义域上是一一映射时才存在。求反函数的步骤是：设 y = f(x)，交换 x 和 y，然后解出 y。对于 f(x) = 2x + 3，有 y = 2x + 3，交换后为 x = 2y + 3，解得 y = (x − 3)/2，因此 f⁻¹(x) = (x − 3)/2。

The graphs of f and f⁻¹ are reflections in the line y = x. This geometric property is useful for checking your answer or finding the inverse graphically. The identities f(f⁻¹(x)) = x and f⁻¹(f(x)) = x hold for all x in the appropriate domains, and they are often tested in proof-style questions.

函数 f 与 f⁻¹ 的图像关于直线 y = x 对称。这一几何性质可用于检验答案或通过图像求反函数。恒等式 f(f⁻¹(x)) = x 和 f⁻¹(f(x)) = x 在相应定义域内恒成立，它们经常出现在证明类题目中。

• Composite functions are evaluated from the inside out: first g, then f. 复合函数由内向外求值：先 g，后 f。
• An inverse exists only if the function is one-to-one on its domain. 只有函数在其定义域上是一一映射时才存在反函数。

4. Quadratic Functions and the Discriminant |

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