Functions (Edexcel A-Level Maths) | 函数（爱德思 A-Level 数学）

📚 Functions (Edexcel A-Level Maths) | 函数（爱德思 A-Level 数学）

Functions are central to A-Level mathematics, linking algebraic expressions to graphical representations and modelling real-world situations. In this revision guide, we cover the key concepts required for Edexcel A-Level Maths, including domain and range, composite and inverse functions, the modulus function, transformations and more.

函数是 A-Level 数学的核心内容，它将代数表达式与图像表示联系起来，并用于模拟现实世界的情形。在本复习指南中，我们将梳理 Edexcel A-Level 数学的重要概念，包括定义域与值域、复合函数与反函数、绝对值函数、图像变换等内容。

1. What is a Function? | 什么是函数？

A function, often denoted as f(x), is a rule that assigns to each input value x exactly one output value y. We write y = f(x). For example, f(x) = x² + 3 is a function. A simple test for whether a graph represents a function is the vertical line test: if any vertical line intersects the graph more than once, it is not a function.

函数通常记作 f(x)，是一个规则，为每个输入值 x 分配恰好一个输出值 y。我们写作 y = f(x)。例如，f(x) = x² + 3 是一个函数。判断一个图像是否表示函数的一个简单方法是垂直线检验：如果任何垂直线与图像相交超过一次，则它不是函数。

All functions have a domain (the set of allowed input values) and a range (the set of possible output values). In Edexcel A-Level, you must be able to define these sets using set notation, such as {x ∈ ℝ : x ≥ 0}.

所有函数都有定义域（允许的输入值集合）和值域（可能的输出值集合）。在爱德思 A-Level 中，你需要能够使用集合表示法来定义这些集合，例如 {x ∈ ℝ : x ≥ 0}。

• f(x) = x² + 5: domain ℝ, range f(x) ≥ 5
• f(x) = 1/(x-3): domain x ≠ 3, range f(x) ≠ 0
• f(x) = √x: domain x ≥ 0, range f(x) ≥ 0

f(x) = x² + 5：定义域 ℝ，值域 f(x) ≥ 5
f(x) = 1/(x-3)：定义域 x ≠ 3，值域 f(x) ≠ 0
f(x) = √x：定义域 x ≥ 0，值域 f(x) ≥ 0

2. Domain and Range | 定义域与值域

The domain of a function is the complete set of all possible input values (x-values) for which the function is defined. Common restrictions include denominators not equal to zero and expressions under a square root being non-negative. For example, f(x) = √(x-2) has domain x ≥ 2, or {x ∈ ℝ : x ≥ 2}.

函数的定义域是使函数有定义的所有可能输入值（x 值）的完整集合。常见的限制包括分母不为零以及平方根下的表达式非负。例如，f(x) = √(x-2) 的定义域为 x ≥ 2，或记作 {x ∈ ℝ : x ≥ 2}。

The range is the set of all output values (y-values) that the function can produce. To find the range, you often need to consider the shape of the graph or complete the square. For f(x) = x² – 4x + 7, by completing the square we get f(x) = (x-2)² + 3, so the minimum value is 3 and the range is f(x) ≥ 3.

值域是函数所能产生的所有输出值（y 值）的集合。为了求出值域，通常需要考虑图像的形状或通过配方法。对于 f(x) = x² – 4x + 7，配方得 f(x) = (x-2)² + 3，因此最小值为 3，值域为 f(x) ≥ 3。

The table above summarises typical domains and ranges for common functions. Notice that for eˣ, the output is always positive, and for ln x, the input must be strictly greater than

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