Functions and Graphs: Domain, Range, and Transformations | 函数与图像：定义域、值域与变换

📚 Functions and Graphs: Domain, Range, and Transformations | 函数与图像：定义域、值域与变换

In this chapter, we explore the fundamental concepts of functions and their graphical representations. We will learn how to determine the domain and range of a function, and how transformations such as translations, reflections, and stretches affect the graph of a function. These ideas are essential for understanding many topics in IGCSE Mathematics and beyond.

在本章中，我们将探讨函数的基本概念及其图形表示。我们将学习如何确定函数的定义域和值域，以及平移、反射和伸缩等变换如何影响函数的图像。这些概念对于理解 IGCSE 数学及其他领域的许多主题至关重要。

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1. What is a Function? | 什么是函数？

A function is a relation that maps each input value (x) to exactly one output value (y). For example, y = 2x + 1 is a function because every x gives a unique y. We often write this as f(x) = 2x + 1.

函数是一种关系，它将每个输入值（x）恰好映射到一个输出值（y）。例如，y = 2x + 1 是一个函数，因为每个 x 都对应唯一的 y。我们通常将其写为 f(x) = 2x + 1。

The set of all possible input values is called the domain, and the set of all possible output values is called the range.

所有可能输入值的集合称为定义域，所有可能输出值的集合称为值域。

Example | 示例: For f(x) = x², the domain is all real numbers, but the range is y ≥ 0 because squaring any real number never gives a negative result.

示例：对于 f(x) = x²，定义域是所有实数，但值域是 y ≥ 0，因为任何实数的平方都不会是负数。

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2. Finding the Domain | 求定义域

The domain is the set of x-values that can be substituted into a function to produce a valid output. We look for values that cause problems, such as division by zero or square roots of negative numbers.

定义域是可以代入函数并产生有效输出的 x 值集合。我们需要找出会导致问题的值，例如分母为零或负数开平方。

• For f(x) = 1/x, x ≠ 0.

  对于 f(x) = 1/x，x ≠ 0。

• For f(x) = √x, x ≥ 0.

  对于 f(x) = √x，x ≥ 0。

• For f(x) = 2x + 3, the domain is all real numbers.

  对于 f(x) = 2x + 3，定义域是所有实数。

Worked Example | 例题: Find the domain of f(x) = √(x – 2).

例题：求 f(x) = √(x – 2) 的定义域。

Since the square root of a negative number is undefined, we need x – 2 ≥ 0, so x ≥ 2. Thus, the domain is x ≥ 2.

因为负数的平方根没有定义，我们需要 x – 2 ≥ 0，即 x ≥ 2。因此，定义域是 x ≥ 2。

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3. Finding the Range | 求值域

The range is the set of all possible output values (y) of a function. To find it, consider what values y can take as x varies over the domain.

值域是函数所有可能输出值（y）的集合。要求值域，考虑当 x 在定义域内变化时，y 可以取哪些值。

Example | 示例: For f(x) = x², since x² ≥ 0 for all x, the range is y ≥ 0.

示例：对于 f(x) = x²，因为对所有 x 都有 x² ≥ 0，所以值域是 y ≥ 0。

For a quadratic function in the form f(x) = ax² + bx + c, we can complete the square to find the maximum or minimum y-value.

对于形如 f(x) = ax² + bx + c 的二次函数，我们可以通过配方法找到最大或最小 y 值。

Worked Example | 例题: Find the range of f(x) = x² – 4x + 5.

例题：求 f(x) = x² – 4x + 5 的值域。

Complete the square: f(x) = (x – 2)² + 1. Since (x – 2)² ≥ 0, the smallest value of f(x) is 1. Therefore, the range is y ≥ 1.

配方：f(x) = (x – 2)² + 1。因为 (x – 2)² ≥ 0，所以 f(x) 的最小值是 1。因此，值域是 y ≥ 1。

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4. Function Notation and Evaluating Functions | 函数记号与求值

Function notation f(x) allows us to evaluate a function at a specific x. For instance, if f(x) = 3x – 2, then f(4) = 3(4) – 2 = 10.

函数记号 f(x) 允许我们在特定的 x 处求函数值。例如，若 f(x) = 3x – 2，则 f(4) = 3(4) – 2 = 10。

Sometimes we encounter composite functions like f(g(x)). This means we first apply g to x, then apply f to the result.

有时我们会遇到复合函数，如 f(g(x))。这意味着我们先对 x 应用 g，再对结果应用 f。

Example | 示例: If f(x) = x + 1 and g(x) = 2x, find f(g(3)).

示例：若 f(x) = x + 1，g(x) = 2x，求 f(g(3))。

First, g(3) = 6. Then f(6) = 7. So f(g(3)) = 7.

首先，g(3) = 6。然后 f(6) = 7。所以 f(g(3)) = 7。

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5. Graphs of Basic Functions | 基本函数图像

Knowing the shape of basic function graphs helps us understand transformations quickly.

了解基本函数的图像形状有助于我们快速理解变换。

Function	Graph shape	Key features
y = x	Straight line through origin	Slope 1
y = x²	Parabola, vertex at origin	Symmetric about y-axis
y = 1/x	Hyperbola	Two branches, asymptotes at x=0, y=0
y = √x	Half-parabola	Starts at origin

Notice that y = x² and y = √x are related, but their domains and ranges differ.

注意 y = x² 和 y = √x 相关，但它们的定义域和值域不同。

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6. Translations | 平移变换

A translation shifts the graph horizontally or vertically without changing its shape. If f(x) is a function, then:

平移变换在不改变图像形状的情况下，将图像水平或垂直移动。若 f(x) 是一个函数，则：

• y = f(x) + a shifts the graph vertically up by a units.

  y = f(x) + a 将图像向上平移 a 个单位。

• y = f(x) – a shifts the graph vertically down by a units.

  y = f(x) – a 将图像向下平移 a 个单位。

• y = f(x – a) shifts the graph horizontally right by a units.

  y = f(x – a) 将图像向右平移 a 个单位。

• y = f(x + a) shifts the graph horizontally left by a units.

  y = f(x + a) 将图像向左平移 a 个单位。

Example | 示例: The graph of y = (x – 3)² is the graph of y = x² shifted right by 3 units.

示例：y = (x – 3)² 的图像是 y = x² 的图像向右平移 3 个单位。

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7. Reflections | 反射变换

Reflections flip the graph across a line. There are two common reflections:

反射变换是将图像关于某条直线翻转。常见的反射有两种：

• y = -f(x) reflects the graph in the x-axis.

  y = -f(x) 将图像关于 x 轴反射。

• y = f(-x) reflects the graph in the y-axis.

  y = f(-x) 将图像关于 y 轴反射。

Example | 示例: If f(x) = x², then y = -x² flips the parabola upside down, while y = (-x)² = x² is unchanged because the parabola is symmetric.

示例：若 f(x) = x²，则 y = -x² 将抛物线翻转，而 y = (-x)² = x² 不变，因为抛物线是对称的。

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8. Stretches and Compressions | 伸缩变换

A stretch multiplies the graph in a particular direction. The rules are:

伸缩变换是在特定方向上将图像放大或缩小。规则如下：

• y = kf(x) vertically stretches the graph if k > 1, and compresses it if 0 < k < 1.

  y = kf(x) 若 k > 1，则纵向拉伸；若 0 < k < 1，则纵向压缩。

• y = f(kx) horizontally compresses the graph if k > 1, and stretches it if 0 < k < 1.

  y = f(kx) 若 k > 1，则横向压缩；若 0 < k < 1，则横向拉伸。

Example | 示例: y = 3x² is the parabola y = x² stretched vertically by a factor of 3. y = (x/2)² = x²/4 is stretched horizontally by a factor of 2.

示例：y = 3x² 是 y = x² 纵向拉伸 3 倍。y = (x/2)² = x²/4 是横向拉伸 2 倍。

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9. Combining Transformations | 组合变换

When multiple transformations are applied in sequence, the order matters. For example, to graph y = 2(x – 1)² + 3, we start with y = x², shift right by 1 unit, stretch vertically by a factor of 2, and shift up by 3 units.

当连续应用多个变换时，顺序很重要。例如，要绘制 y = 2(x – 1)² + 3，我们从 y = x² 开始，向右平移 1 个单位，纵向拉伸 2 倍，再向上平移 3 个单位。

Worked Example | 例题: Describe the transformations from y = 1/x to y = 1/(x – 2) + 4.

例题：描述从 y = 1/x 到 y = 1/(x – 2) + 4 的变换。

First, y = 1/(x – 2) is a translation of y = 1/x by 2 units to the right. Then y = 1/(x – 2) + 4 is a translation by 4 units upward. So the graph shifts right 2 and up 4.

首先，y = 1/(x – 2) 是 y = 1/x 向右平移 2 个单位。然后 y = 1/(x – 2) + 4 是向上平移 4 个单位。因此，图像向右平移 2 个单位，再向上平移 4 个单位。

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10. Reading Graphs | 读图

To find the domain and range from a graph, look at the x-values covered by the curve and the y-values covered by the curve.

要从图像中求定义域和值域，需观察曲线覆盖的 x 值范围和 y 值范围。

For example

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