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Functions: Concepts, Graphs, and Applications | 函数：概念、图像与应用

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📚 Functions: Concepts, Graphs, and Applications | 函数：概念、图像与应用

Functions are one of the most fundamental concepts in mathematics, forming the backbone of algebra, calculus, and applied mathematics. In this revision guide, we will explore the definition of functions, their graphical representations, and their real-world applications, with a focus on the skills you need for A-Level examinations.

函数是数学中最基本的概念之一，构成了代数、微积分和应用数学的基石。在本复习指南中，我们将探讨函数的定义、图像表示及其在实际世界中的应用，重点关注A-Level考试所需的技能。

1. The Concept of a Function | 函数的概念

A function is a relation between a set of inputs and a set of permissible outputs, where each input is related to exactly one output. We write this as \( f : A \rightarrow B \), or more simply, \( y = f(x) \), where \( x \) is the independent variable and \( y \) is the dependent variable.

函数是输入集合与许可输出集合之间的一种关系，其中每个输入恰好对应一个输出。我们将其记为 \( f : A \rightarrow B \)，或者更简单地写作 \( y = f(x) \)，其中 \( x \) 是自变量，\( y \) 是因变量。

For a relation to be a function, it must satisfy the vertical line test: any vertical line drawn through the graph must intersect the curve at most once. This ensures that each \( x \)-value has only one corresponding \( y \)-value.

一个关系要成为函数，必须满足垂直线检验法：任意一条垂直于 \( x \)-轴的直线与曲线的交点至多一个。这保证了每个 \( x \)-值只有一个对应的 \( y \)-值。

Common examples of functions include:

常见函数的例子包括：

  • Linear functions: \( f(x) = mx + c \) | 线性函数：\( f(x) = mx + c \)
  • Quadratic functions: \( f(x) = ax² + bx + c \) | 二次函数：\( f(x) = ax² + bx + c \)
  • Cubic functions: \( f(x) = ax³ + bx² + cx + d \) | 三次函数：\( f(x) = ax³ + bx² + cx + d \)
  • Exponential functions: \( f(x) = a^x \) | 指数函数：\( f(x) = a^x \)
  • Logarithmic functions: \( f(x) = \log_a x \) | 对数函数：\( f(x) = \log_a x \)

2. Domain and Range | 定义域与值域

The domain of a function is the complete set of possible input values (\( x \)-values) for which the function is defined. The range is the complete set of all possible output values (\( y \)-values) that the function can produce.

定义域是函数有定义的所有可能的输入值（\( x \)-值）的完整集合。值域是函数能够产生的所有可能的输出值（\( y \)-值）的完整集合。

When determining the domain, watch out for these restrictions:

求定义域时需要特别注意以下限制：

  • Denominators cannot be zero — exclude values where the denominator equals zero. | 分母不能为零——排除使分母等于零的数值。
  • Expressions under a square root (or any even root) must be non-negative. | 平方根（或任何偶次根）下的表达式必须非负。
  • Arguments of logarithms must be positive. | 对数的真数必须为正。

Example: Find the domain and range of \( f(x) = \sqrt{x – 3} \).

示例：求 \( f(x) = \sqrt{x – 3} \) 的定义域和值域。

For the domain, we require \( x – 3 \geq 0 \), so \( x \geq 3 \). Thus, the domain is [3, ∞). Since the square root produces non-negative outputs, the range is [0, ∞).

对于定义域，我们需要 \( x – 3 \geq 0 \)，即 \( x \geq 3 \)。因此，定义域为 [3, ∞)。由于平方根产生非负的输出值，值域为 [0, ∞)。

3. Function Notation and Composite Functions | 函数符号与复合函数

Function notation \( f(x) \) allows us to evaluate functions at specific points. For example, if \( f(x) = 2x + 1 \), then \( f(3) = 2(3) + 1 = 7 \).

函数符号 \( f(x) \) 使我们可以计算函数在特定点的值。例如，若 \( f(x) = 2x + 1 \)，则 \( f(3) = 2(3) + 1 = 7 \)。

A composite function is formed when one function is applied to the result of another. The composition of \( f \) and \( g \), written as \( (f \circ g)(x) = f(g(x)) \), means we apply \( g \) first, then \( f \).

复合函数是指将一个函数应用于另一个函数的结果而构成的新函数。\( f \) 与 \( g \) 的复合记作 \( (f \circ g)(x) = f(g(x)) \)，表示先应用 \( g \)，再应用 \( f \)。

Example: Given \( f(x) = x² \) and \( g(x) = x + 2 \), find \( f(g(x)) \) and \( g(f(x)) \).

示例：已知 \( f(x) = x² \)，\( g(x) = x + 2 \)，求 \( f(g(x)) \) 和 \( g(f(x)) \)。

\( f(g(x)) = f(x + 2) = (x + 2)² = x² + 4x + 4 \)

\( g(f(x)) = g(x²) = x² + 2 \)

Note: \( f(g(x)) \neq g(f(x)) \) in general | 注意：一般情况下 \( f(g(x)) \neq g(f(x)) \)

This demonstrates that composition is not commutative — the order in which you apply functions matters.

这表明复合运算不满足交换律——函数的应用顺序很重要。

4. Inverse Functions | 反函数

An inverse function reverses the effect of the original function. If \( f(a) = b \), then the inverse function \( f^{-1}(b) = a \). The domain of \( f^{-1} \) is the range of \( f \), and vice versa.

反函数逆转原函数的作用。若 \( f(a) = b \)，则反函数 \( f^{-1}(b) = a \)。\( f^{-1} \) 的定义域是 \( f \) 的值域，反之亦然。

For a function to have an inverse, it must be one-to-one (injective) — meaning each output corresponds to exactly one input. This can be verified using the horizontal line test.

一个函数要有反函数，必须是一一对应的（单射）——即每个输出值恰好对应一个输入值。这可以通过水平线检验法来验证。

Finding the Inverse:

求反函数的步骤：

  • Replace \( f(x) \) with \( y \). | 将 \( f(x) \) 替换为 \( y \)。
  • Swap \( x \) and \( y \). | 交换 \( x \) 和 \( y \)。
  • Solve for \( y \). | 解出 \( y \)。
  • Replace \( y \) with \( f^{-1}(x) \). | 将 \( y \) 替换为 \( f^{-1}(x) \)。

Example: Find the inverse of \( f(x) = 3x – 5 \).

示例：求 \( f(x) = 3x – 5 \) 的反函数。

Let \( y = 3x – 5 \). Swap: \( x = 3y – 5 \). Solving: \( 3y = x + 5 \), so \( y = \frac{x + 5}{3} \). Therefore \( f^{-1}(x) = \frac{x + 5}{3} \).

设 \( y = 3x – 5 \)。交换后得 \( x = 3y – 5 \)。解出：\( 3y = x + 5 \)，所以 \( y = \frac{x + 5}{3} \)。因此 \( f^{-1}(x) = \frac{x + 5}{3} \)。

Graphically, the inverse function is the reflection of the original function across the line \( y = x \).

从图像上看，反函数是原函数关于直线 \( y = x \) 的镜像对称。

5. Graph Transformations | 图像变换

Graph transformations allow us to sketch complex functions by applying simple transformations to known parent functions. The key transformations are:

图像变换使我们能够通过对已知的基本函数应用简单的变换来绘制复杂函数的草图。关键的变换包括：

Transformation | 变换 Function | 函数 Effect | 效果
Vertical translation | 垂直平移 \( f(x) + a \) Shift up by \( a \) if \( a > 0 \), down if \( a < 0 \). | 当 \( a > 0 \) 时向上平移 \( a \)，\( a < 0 \) 时向下平移。
Horizontal translation | 水平平移 \( f(x – a) \) Shift right by \( a \) if \( a > 0 \), left if \( a < 0 \). | 当 \( a > 0 \) 时向右平移 \( a \)，\( a < 0 \) 时向左平移。
Vertical stretch | 垂直伸缩 \( a f(x) \) Stretch vertically by factor \( a \). | 垂直方向伸缩 \( a \) 倍。
Horizontal stretch | 水平伸缩 \( f(ax) \) Compress horizontally by factor \( \frac{1}{a} \). | 水平方向压缩 \( \frac{1}{a} \) 倍。
Reflection in x-axis | 关于 x 轴对称 \( -f(x) \) Flip across the x-axis. | 沿 x 轴翻转。
Reflection in y-axis | 关于 y 轴对称 \( f(-x) \) Flip across the y-axis. | 沿 y 轴翻转。

Worked Example: Describe the transformations needed to obtain \( y = 2(x – 1)² + 3 \) from \( y = x² \).

例题：描述从 \( y = x² \) 得到 \( y = 2(x – 1)² + 3 \) 所需的变换。

  • Shift right by 1 unit: \( y = (x – 1)² \). | 向右平移 1 个单位：\( y = (x – 1)² \)。
  • Vertical stretch by factor 2: \( y = 2(x – 1)² \). | 垂直伸缩 2 倍：\( y = 2(x – 1)² \)。
  • Shift up by 3 units: \( y = 2(x – 1)² + 3 \). | 向上平移 3 个单位：\( y = 2(x – 1)² + 3 \)。

Remember that the order of transformations matters. Horizontal shifts and stretches must be applied to the input \( x \) before vertical transformations.

请注意变换的顺序很重要。水平平移和伸缩必须先作用于输入 \( x \)，然后再进行垂直变换。

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

Exponential functions take the form \( f(x) = a^x \) where \( a > 0 \) and \( a \neq 1 \). Logarithms are the inverse of exponential functions: if \( a^x = y \), then \( \log_a y = x \).

指数函数的形式为 \( f(x) = a^x \)，其中 \( a > 0 \) 且 \( a \neq 1 \)。对数是指数函数的反函数：若 \( a^x = y \)，则 \( \log_a y = x \)。

Key properties of logarithms you must know:

对数必须掌握的关键性质：

  • \( \log_a (xy) = \log_a x + \log_a y \) | 积的对数等于对数之和
  • \( \log_a \left( \frac{x}{y} \right) = \log_a x – \log_a y \) | 商的对数等于对数之差
  • \( \log_a (x^n) = n \log_a x \) | 幂的对数等于指数乘对数
  • \( \log_a 1 = 0 \) and \( \log_a a = 1 \) | \( \log_a 1 = 0 \)，\( \log_a a = 1 \)
  • Change of base: \( \log_a x = \frac{\log_b x}{\log_b a} \) | 换底公式：\( \log_a x = \frac{\log_b x}{\log_b a} \)

Example: Solve \( 5^x = 20 \).

示例：解方程 \( 5^x = 20 \)。

Taking logs of both sides: \( \log 5^x = \log 20 \), so \( x \log 5 = \log 20 \). Therefore:

两边同时取对数：\( \log 5^x = \log 20 \)，所以 \( x \log 5 = \log 20 \)。因此：

\( x = \frac{\log 20}{\log 5} \approx \frac{1.3010}{0.6987} \approx 1.861 \)

The graph of \( y = a^x \) passes through (0, 1), increases when \( a > 1 \), and approaches the x-axis asymptotically as \( x \rightarrow -\infty \). The logarithmic graph \( y = \log_a x \) is its reflection across \( y = x \), passing through (1, 0).

\( y = a^x \) 的图像过点 (0, 1)，当 \( a > 1 \) 时递增，并且当 \( x \rightarrow -\infty \) 时渐近逼近 x-轴。对数函数 \( y = \log_a x \) 的图像是其关于 \( y = x \) 的镜像，过点 (1, 0)。

7. Trigonometric Functions | 三角函数

The three main trigonometric functions — sine, cosine, and tangent — are periodic functions with distinct graph shapes. Understanding their graphs is essential for solving equations and modelling periodic phenomena.

三个主要的三角函数——正弦、余弦和正切——是具有不同图像形状的周期函数。理解它们的图像对于解方程和建立周期现象的模型至关重要。

Function | 函数 Period | 周期 Range | 值域 Key Points | 关键点
\( y = \sin x \) \( 360° \) or \( 2\pi \) \( -1 \leq y \leq 1 \) \( (0,0), (90°,1), (180°,0), (270°,-1), (360°,0) \)
\( y = \cos x \) \( 360° \) or \( 2\pi \) \( -1 \leq y \leq 1 \) \( (0,1), (90°,0), (180°,-1), (270°,0), (360°,1) \)
\( y = \tan x \) \( 180° \) or \( \pi \) All real numbers | 全体实数 \( (0,0), (45°,1), (135°,-1), (180°,0) \); asymptotes at \( x = 90° + 180°n \) | 在 \( x = 90° + 180°n \) 处有渐近线

Key identities to memorise:

必须记忆的关键恒等式：

\( \sin²\theta + \cos²\theta = 1 \),    \( \tan\theta = \frac{\sin\theta}{\cos\theta} \)

These identities are essential for simplifying expressions and proving other relationships. The general solutions of trigonometric equations must account for periodicity:

这些恒等式对于化简表达式和证明其他关系至关重要。三角方程的通解必须考虑周期性：

If \( \sin\theta = c \), then \( \theta = \sin^{-1}(c) + 360°n \) or \( 180° – \sin^{-1}(c) + 360°n \)

若 \( \sin\theta = c \)，则 \( \theta = \sin^{-1}(c) + 360°n \) 或 \( 180° – \sin^{-1}(c) + 360°n \)

8. Transformations of Trigonometric Graphs | 三角函数图像的变换

Trigonometric graphs can be transformed using the general rules from Section 5. However, the parameters in \( y = a\sin(bx + c) + d \) have specific meanings that you should be able to identify immediately:

三角函数图像可以使用第5节中的一般规则进行变换。然而，在 \( y = a\sin(bx + c) + d \) 中，各个参数有着你需要能够立即识别出的特定含义：

  • \( a \) — amplitude | 振幅：vertical stretch factor, maximum displacement from the centre line. | 垂直伸缩因子，距离中心线的最大位移。
  • \( b \) — period = \( \frac{360°}{b} \) or \( \frac{2\pi}{b} \) | 周期 = \( \frac{360°}{b} \) 或 \( \frac{2\pi}{b} \)。
  • \( c \) — horizontal shift: the graph shifts left by \( \frac{c}{b} \) when \( c > 0 \) | 水平平移：当 \( c > 0 \) 时，图像向左平移 \( \frac{c}{b} \)。
  • \( d \) — vertical shift: moves the centre line up or down. | 垂直平移：将中心线向上或向下移动。

Example: State the amplitude, period, and phase shift of \( y = 3\sin(2x – 60°) + 1 \).

示例：写出 \( y = 3\sin(2x – 60°) + 1 \) 的振幅、周期和相位平移。

  • Amplitude | 振幅：\( 3 \)
  • Period | 周期：\( \frac{360°}{2} = 180° \)
  • Phase shift | 相位平移：\( 2x – 60° = 0 \Rightarrow x = 30° \), so shift right by \( 30° \). | \( 2x – 60° = 0 \Rightarrow x = 30° \)，即向右平移 \( 30° \)。
  • Vertical shift | 垂直平移：\( +1 \), centre line is \( y = 1 \). | \( +1 \)，中心线为 \( y = 1 \)。

9. Applications: Modelling with Functions | 应用：用函数建模

Functions are powerful tools for modelling real-world situations. Here are some classic applications that frequently appear in examinations:

函数是模拟现实世界情况的强大工具。以下是一些在考试中经常出现的经典应用：

1. Linear Models — Cost and Revenue | 线性模型——成本与收入

If a company has fixed costs of £200 and produces items at £5 each, the total cost function is \( C(x) = 5x + 200 \), where \( x \) is the number of items. The revenue from selling at £12 each is \( R(x) = 12x \). The break-even point occurs where \( C(x) = R(x) \).

如果一家公司有 200 英镑的固定成本，每生产一件产品花费 5 英镑，那么总成本函数为 \( C(x) = 5x + 200 \)，其中 \( x \) 是产品数量。以每件 12 英镑的价格销售的收入为 \( R(x) = 12x \)。盈亏平衡点出现在 \( C(x) = R(x) \) 处。

\( 5x + 200 = 12x \Rightarrow 7x = 200 \Rightarrow x \approx 28.6 \), so break-even at approximately 29 items

\( 5x + 200 = 12x \Rightarrow 7x = 200 \Rightarrow x \approx 28.6 \)，即大约生产 29 件时达到盈亏平衡

2. Exponential Models — Population Growth and Radioactive Decay | 指数模型——人口增长与放射性衰变

Exponential growth follows \( y = A(1 + r)^t \), where \( A \) is the initial amount, \( r \) is the growth rate, and \( t \) is time. Radioactive decay follows \( y = Ae^{-kt} \), where \( k \) is the decay constant.

指数增长遵循 \( y = A(1 + r)^t \)，其中 \( A \) 是初始量，\( r \) 是增长率，\( t \) 是时间。放射性衰变遵循 \( y = Ae^{-kt} \)，其中 \( k \) 是衰变常数。

3. Quadratic Models — Projectile Motion | 二次模型——抛体运动

The height of a projectile follows \( h(t) = -4.9t² + v_0t + h_0 \) (in metres, with \( g \approx 9.8 \) m/s²). The vertex of this parabola gives the maximum height, and the roots give the flight time.

抛体的高度遵循 \( h(t) = -4.9t² + v_0t + h_0 \)（单位为米，\( g \approx 9.8 \) m/s²）。该抛物线的顶点给出最大高度，而根给出飞行时间。

10. Solving Equations Graphically | 图像法解方程

Graphs provide a visual method for solving equations. To solve \( f(x) = g(x) \), plot both functions on the same axes and find the \( x \)-coordinates of their intersection points.

图像提供了一种解方程的视觉方法。要求解 \( f(x) = g(x) \)，只需在同一坐标系中绘制两个函数的图像，并找到它们交点的 \( x \)-坐标。

To solve \( f(x) = 0 \), find the \( x \)-intercepts (roots) of the graph. The number of real roots can be determined by the number of times the graph crosses the x-axis.

要求解 \( f(x) = 0 \)，找出图像与 \( x \)-轴的交点（根）。实根的个数可以通过图像与 \( x \)-轴相交的次数来确定。

Example: How many real roots does \( x³ – 3x + 1 = 0 \) have?

示例：方程 \( x³ – 3x + 1 = 0 \) 有多少个实根？

Consider \( f(x) = x³ – 3x + 1 \). We evaluate:

考虑 \( f(x) = x³ – 3x + 1 \)。我们计算：

  • \( f(-2) = -8 + 6 + 1 = -1 \) (negative | 负值)
  • \( f(-1) = -1 + 3 + 1 = 3 \) (positive | 正值)
  • \( f(0) = 1 \) (positive | 正值)
  • \( f(1) = 1 – 3 + 1 = -1 \) (negative | 负值)
  • \( f(2) = 8 – 6 + 1 = 3 \) (positive | 正值)

By the Intermediate Value Theorem, there is a root between \(-2\) and \(-1\), another between \(0\) and \(1\), and a third between \(1\) and \(2\). Since a cubic has at most three real roots, there are exactly three real roots.

由介值定理可知，在 \(-2\) 和 \(-1\) 之间有一个根，在 \(0\) 和 \(1\) 之间有另一个根，在 \(1\) 和 \(2\) 之间有第三个根。由于三次函数至多有三个实根，因此恰好有三个实根。

11. The Sign of a Function and Inequalities | 函数的符号与不等式

Knowing where a function is positive or negative is essential for solving inequalities. For a quadratic \( f(x) = ax² + bx + c \), the sign depends on the roots and the coefficient \( a \).

了解函数在何处为正或为负对于解不等式至关重要。对于二次函数 \( f(x) = ax² + bx + c \) 而言，其符号取决于根和系数 \( a \)。

If \( a > 0 \), the graph is a “smile” — positive outside the roots and negative between them. If \( a < 0 \), the graph is a "frown" — the reverse.

若 \( a > 0 \)，图像呈”微笑”形——在两根之外为正，在两根之间为负。若 \( a < 0 \)，图像呈"皱眉"形——情况相反。

Example: Solve \( x² – x – 6 > 0 \).

示例：解不等式 \( x² – x – 6 > 0 \)。

Factorising: \( (x – 3)(x + 2) > 0 \). The roots are \( x = -2 \) and \( x = 3 \). Since \( a = 1 > 0 \), the solution is \( x < -2 \) or \( x > 3 \).

因式分解：\( (x – 3)(x + 2) > 0 \)。根为 \( x = -2 \) 和 \( x = 3 \)。由于 \( a = 1 > 0 \)，解为 \( x < -2 \) 或 \( x > 3 \)。

For rational functions, pay attention to values that make the denominator zero, as these create asymptotes and divide the number line into intervals for sign analysis.

对于有理函数，注意使分母为零的值，因为这些值会产生渐近线并将数轴划分为用于符号分析的区间。

12. Exam Tips and Common Mistakes | 考试技巧与常见错误

Common mistakes students make in functions questions:

学生在函数题目中的常见错误：

  • Confusing the notation \( f^{-1}(x) \) with \( \frac{1}{f(x)} \). The inverse function is NOT the reciprocal! | 混淆 \( f^{-1}(x) \) 与 \( \frac{1}{f(x)} \) 的记号。反函数不是倒数！
  • Forgetting to restrict the domain when finding an inverse of a non-one-to-one function. | 在求非一一对应函数的反函数时，忘记限制定义域。
  • Applying transformations in the wrong order. | 以错误的顺序应用变换。
  • When solving \( \log_a x + \log_a (x – 2) \), forgetting to check that solutions satisfy the original domain. | 在解 \( \log_a x + \log_a (x – 2) \) 时，忘记检验解是否满足原始定义域。
  • Mixing up horizontal and vertical transformations — \( f(x + a) \) shifts left, not right. | 混淆水平和垂直变换——\( f(x + a) \) 是向左平移，不是向右。

Key strategies for success:

取得成功的关键策略：

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