📚 IGCSE WJEC Maths: Functions Revision | IGCSE WJEC 数学：函数考点精讲
Mastering functions is vital for success in the IGCSE WJEC Mathematics examination. From understanding basic function notation to working confidently with composite and inverse functions, these skills form the foundation for many advanced topics. This detailed revision guide explains every key concept you need, with clear examples and bilingual summaries to strengthen your understanding.
熟练掌握函数是征服 IGCSE WJEC 数学考试的关键。从理解基础的函数符号到灵活运用复合函数与反函数，这些技能为许多高阶段主题奠定了基础。本篇详细的考点精讲，为你梳理每一个必会概念，辅以清晰的范例与双语总结，帮助你牢固掌握知识。
1. Function Notation and Basics | 函数符号与基础概念
maps each input value to exactly one output value. We often use the notation f(x) (read as 'f of x' ction f when the input is x. For example, f(x) = 2x + 3 means multiply x by 2, then add 3.
函数是一种规则，将每一个输入值唯一地对应到一个输出值。我们常用 f(x)（读作 f of x）表示当输入为 x 时函数 f 的输出。例如，f(x) = 2x + 3 表示将 x 乘以 2，再加 3。
(4) = 4² - 5 = 16 - 5 = 11. Always replace x carefully and follow the order of operations.
你可以将具体的数值代入 x 来计算函数值。若 f(x) = x² - 5，则 f(4) = 4² - 5 = 16 - 5 = 11。替换 x 时务必仔细，并遵循运算优先级。
- Function machine idea: input → rule → output
- 函数机器思路：输入 → 规则 → 输出
- f(x) = expression; the letter f names the function, x is the variable
- f(x) = 表达式；字母 f 表示函数名，x 为变量
- Evaluating means replacing x with a number or another algebraic expression
- 求值指用数字或另一个代数式替换 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 set of all output values (f(x) values) the function can produce. For most IGCSE problems, the domain might be all real numbers unless a restriction is given or makes mathematical sense.
函数的定义域是指能使函数有意义的所有可能输入值（x 值）的集合。值域则是函数所能产生的所有输出值（f(x) 值）的集合。在大多数 IGCSE 题目中，若没有特殊限制或数学上无意义，定义域通常为全体实数。
vision by zero is not allowed, so the domain is all real numbers except 2. The range can then be found by considering which output values are impossible; here f(x) can never equal 0.
例如，函数 f(x) = 1/(x - 2) 在 x = 2 时无定义，因为分母不能为零，因此定义域为除了 2 以外的全体实数。随后通过考虑哪些输出值不可能出现来求值域；此处 f(x) 永远不可能等于 0。
- Domain: look out for denominators not zero, square roots of non-negative numbers if real-valued functions
- 定义域：注意分母不为零，若为实值函数，平方根下的表达式必须非负
- Range: sketch a graph or use algebraic reasoning; extremes or asymptotes give clues
- 值域：画草图或使用代数推理；极值与渐近线提供线索
- Notation: can use sets or inequalities, e.g. x ∈ ℝ, x ≠ 2, or x > 0
- 表示法：可使用集合或不等式，例如 x ∈ ℝ, x ≠ 2 或 x > 0
3. Composite Functions | 复合函数
A composite function applies one function to the result of another. The notation fg(x) means first apply g to x, then apply f to the result. This is sometimes written as f(g(x)). It is crucial to work from the inside out.
复合函数是指将一个函数作用于另一个函数的结果。记号 fg(x) 表示先对 x 使用函数 g，再将所得结果代入 f。有时也写作 f(g(x))。关键要由内向外运算。
²) = 3x² - 1. Notice that gf(x) would be (3x - 1)², which is usually different from fg(x).
设 f(x) = 3x - 1，g(x) = x²。那么 fg(2) = f(g(2)) = f(4) = 3(4) - 1 = 11。其代数形式为 fg(x) = f(x²) = 3x² - 1。注意 gf(x) 将是 (3x - 1)²，它通常与 fg(x) 不同。
- fg(x) = f(g(x)): apply g first, then f
- fg(x) = f(g(x))：先用 g，再用 f
- gf(x) means apply f, then g; order matters greatly
- gf(x) 表示先用 f，再用 g；顺序极为重要
- Work methodically, especially when domain issues arise with composed functions
- 有条理地计算，尤其当复合函数涉及定义域限制时
4. Inverse Functions | 反函数
an inverse, it must be one-to-one (each output comes from exactly one input). The inverse 'undoes' the original function: f⁻¹(f(x)) = x.
函数 f 的反函数记作 f⁻¹(x)，它反转 f 的作用。函数必须有单射性（每个输出值恰好来自一个输入值）才能拥有反函数。反函数“撤销”原函数的效果：f⁻¹(f(x)) = x。
et y = (x - 3)/2. Swap: x = (y - 3)/2. Solve: 2x = y - 3 → y = 2x + 3, so f⁻¹(x) = 2x + 3.
代数上求反函数的方法：令 y = f(x)，交换 x 与 y，再解出 y。例如，若 f(x) = (x - 3)/2，设 y = (x - 3)/2。交换后：x = (y - 3)/2。解得：2x = y - 3 → y = 2x + 3，因此 f⁻¹(x) = 2x + 3。
- One-to-one check: using horizontal line test on graph; restrict domain if not one-to-one
- 单射性检验：在图像上使用水平线测试；若并非单射，可限制定义域
- Swap and solve method is standard; remember to write answer in function notation
- 交换再解的方法是标准做法；记得用函数符号书写答案
- Graph of inverse is reflection of original in the line y = x
- 反函数的图像是原函数关于直线 y = x 的对称图形
5. Graphing Functions | 函数图像的绘制
IGCSE candidates must be able to sketch and interpret graphs of various functions: linear, quadratic, cubic, reciprocal, and exponential. Key features include intercepts, turning points, asymptotes, and general shape. Accurate plotting can be supported by constructing a table of values.
IGCSE 考生必须能绘制与解读各类函数的图像，包括一次函数、二次函数、三次函数、倒数函数和指数函数。关键特征包括截距、转折点、渐近线以及整体形状。通过列表取值可以辅助精确绘图。
- Linear: y = mx + c, straight line with gradient m and y-intercept c
- 一次：y = mx + c，斜率为 m、y 轴截距为 c 的直线
- Quadratic: y = ax² + bx + c, parabola; positive a gives ∪ shape, negative a gives ∩ shape
- 二次：y = ax² + bx + c，抛物线；a 为正时开口向上，a 为负时开口向下
- Cubic: y = ax³ + bx² + cx + d, 'S' shaped, can have one or two turning points
- 三次：y = ax³ + bx² + cx + d，呈 S 形，可有一到两个转折点
- Reciprocal: y = k/x, two branches with asymptotes at x=0 and y=0
- 倒数：y = k/x，两条分支，以 x=0 和 y=0 为渐近线
- Exponential: y = ax (a>0, a≠1), rapid growth or decay, horizontal asymptote y=0
- 指数：y = aˣ（a>0, a≠1），快速增长或衰减，水平渐近线 y=0
6. Transformations of Functions | 函数图像的变换
Transformations allow you to shift, stretch, or reflect the graph of a function. Understanding how modifications to the function equation affect the graph is a core skill. The four main types are translation, reflection, stretch, and (less commonly at IGCSE) rotation. Only the first three are typically examined.
通过变换，你可以对函数图像进行平移、拉伸或对称操作。理解函数表达式的变化如何影响图像是一项核心技能。四种主要类型是平移、对称、拉伸，以及（IGCSE 较少见的）旋转。通常只考查前三种。
If f(x) is the original function, then f(x - a) translates the graph a units to the right, f(x) + a translates it a units up. f(-x) reflects the graph in the y-axis, -f(x) reflects it in the x-axis. For stretches, a f(x) with a > 1 stretches vertically by factor a, while f(ax) with a > 1 compresses horizontally by factor 1/a.
若原函数为 f(x)，则 f(x - a) 将图像向右平移 a 个单位，f(x) + a 将图像向上平移 a 个单位。f(-x) 将图像关于 y 轴对称，-f(x) 则是关于 x 轴对称。对于拉伸，a f(x)（a > 1）使图像沿纵向拉伸 a 倍，而 f(ax)（a > 1）使图像沿横向压缩为原来的 1/a。
- Translation by vector (h, k): y = f(x - h) + k
- 按向量 (h, k) 平移：y = f(x - h) + k
- Reflection in x-axis: y = -f(x); in y-axis: y = f(-x)
- 关于 x 轴对称：y = -f(x)；关于 y 轴对称：y = f(-x)
- Vertical stretch factor a: y = a f(x); horizontal stretch factor 1/a: y = f(ax)
- 纵向拉伸 a 倍：y = a f(x)；横向拉伸 1/a 倍：y = f(ax)
- Combining transformations: apply changes closest to x first (horizontal) then outside changes (vertical)
- 组合变换：先执行最靠近 x 的变化（横向），再执行外层变化（纵向）
7. Solving Equations Using Functions | 利用函数解方程
ere the graphs of y = f(x) and y = g(x) intersect. This visual approach is particularly useful when algebraic methods are cumbersome.
许多方程可以利用函数的观点来解决。例如，从图像上解 f(x) = g(x) 即为寻找 y = f(x) 与 y = g(x) 的图像交点的 x 坐标。当代数方法繁琐时，这种直观方法尤其有效。
finding where the curve cuts the x-axis (roots). Similarly, you can solve equations like 2ˣ = x + 1 by sketching y = 2ˣ and y = x + 1, then estimating the intersection point(s).
二次方程可通过绘制 y = ax² + bx + c 的图像，并找出抛物线与 x 轴的交点（即根）来求解。类似地，绘制 y = 2ˣ 和 y = x + 1 的草图并估算交点，即可解出 2ˣ = x + 1 这样的方程。
- Graphical solutions give approximate answers; often you'll be asked to read values to 1 decimal place
- 图像解法给出近似答案；题目通常要求读数精确至一位小数
- You can also use function graphs to solve inequalities, e.g. f(x) > g(x) is where f's graph is above g's
- 函数图像也可用于解不等式，如 f(x) > g(x) 对应 f 图像位于 g 图像上方的区域
8. Real-World Applications of Functions | 函数的实际应用
Functions model real-life situations: cost functions, distance-time graphs, temperature variations, and profit maximisation problems often appear in WJEC exams. Translating a written problem into a function is the first step; interpreting the result in context is the final step.
函数可以模拟现实生活场景，如成本函数、路程-时间图像、温度变化和利润最大化问题，这些在 WJEC 考试中经常出现。将文字题转化为函数是第一步；结合情境解读结果则是最后一步。
For example, a company's profit P (in £ points (solving P(n) = 0). Always check that answers make practical sense (e.g. n cannot be negative).
例如，某公司利润 P（英镑）与售出商品数量 n 的关系为 P(n) = -2n² + 120n - 1000。你可能需要求最大利润（二次函数顶点）或盈亏平衡点（解 P(n)=0）。始终要检查答案在实际情境中是否合理（例如 n 不能为负）。
- Identify variables: state clearly what x and y represent
- 明确变量：清晰说明 x 和 y 分别代表什么
- Use maximum/minimum of quadratics by completing the square or using vertex formula
- 通过配方法或顶点公式求二次函数的最值
- Range of function may be limited by practical constraints, e.g. domain x ≥ 0
- 函数的值域可能受实际限制，例如定义域 x ≥ 0
9. Common Mistakes and Exam Tips | 常见错误与应试技巧
Students often confuse inverse and composite operations, or mix up the order in composite functions. Another common pitfall is incorrectly applying transformations, especially horizontal shifts (f(x + 2) moves left, not right). Misreading the domain can lead to extraneous answers.
学生常混淆反函数与复合运算，或在复合函数中搞错顺序。另一个常见陷阱是错误地应用变换，特别是水平平移（f(x + 2) 向左移，而非向右）。误读定义域会引出多余的答案。
a question asks for f⁻¹(x), check your result by verifying f(f⁻¹(x)) = x. Manage time by practicing with past papers under timed conditions.
考试中务必展现清晰的解题步骤：写出函数表达式，逐步代入，绘制图像时标注坐标轴。若题目要求求 f⁻¹(x)，通过验证 f(f⁻¹(x)) = x 来检查结果。在限时条件下练习历年真题以掌控时间。
- Double-check order: fg(x) not gf(x)
- 仔细核对顺序：是 fg(x) 不是 gf(x)
- Watch for brackets when substituting; especially important with negatives
- 代入时注意括号；涉及负数时尤为重要
- Graph sketches need not be perfect but must show key features (intercepts, turning points, asymptotes)
- 草图不必完美，但必须体现关键特征（截距、转折点、渐近线）
- Learn standard transformation rules by heart to save time
- 熟记标准变换规则以节省时间
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