Composite Functions and Graph Transformations | 复合函数与图像变换

📚 Composite Functions and Graph Transformations | 复合函数与图像变换

In Edexcel A-Level Mathematics, composite functions and graph transformations are core tools for understanding how algebraic expressions and curves behave. This article covers the definitions, notation, inverse functions, combined transformations, and common exam pitfalls. You will also see how these ideas connect to real questions where order, domain, and graphical evidence matter.

在 Edexcel A-Level 数学中，复合函数与图像变换是理解代数式与曲线行为的核心工具。本文涵盖定义、记号、反函数、组合变换以及常见考试失分点。你还会看到这些思想如何与真实考题联系起来，其中顺序、定义域和图像证据都非常关键。

1. What Is a Composite Function? | 什么是复合函数

A composite function is formed when one function is applied and then a second function is applied to the first result. If f and g are functions, the composite f∘g means f(g(x)). First apply g to x, then apply f to the output of g. The order of application is read from right to left in the notation f∘g.

当一个函数作用后，再把另一个函数作用于第一次的结果上，就形成了复合函数。若 f 和 g 是函数，复合函数 f∘g 表示 f(g(x))。先把 g 作用于 x，再把 f 作用于 g 的输出。在记号 f∘g 中，作用的顺序是从右向左读。

2. Notation and Evaluation | 符号与求值

The expression f∘g is read as ‘f composed with g’. Do not confuse it with multiplication. For example, if f(x) = x + 2 and g(x) = 3x, then f(g(x)) = 3x + 2, but g(f(x)) = 3(x + 2) = 3x + 6. The two results are different because order matters in composition.

表达式 f∘g 读作“f 复合 g”。不要把它与乘法混淆。例如，若 f(x) = x + 2 且 g(x) = 3x，则 f(g(x)) = 3x + 2，但 g(f(x)) = 3(x + 2) = 3x + 6。两个结果不同，因为复合中顺序很重要。

3. Domain and Range of Composite Functions | 复合函数的定义域与值域

For f(g(x)) to be defined, g(x) must lie in the domain of f, and x must lie in the domain of g. The domain of f∘g is therefore a subset of the domain of g. The range of f∘g is found by applying f to the range of g, but restricted to the actual domain. Always check for excluded values such as division by zero or negative square roots.

要使 f(g(x)) 有定义，g(x) 必须落在 f 的定义域内，且 x 必须落在 g 的定义域内。因此 f∘g 的定义域是 g 的定义域的一个子集。f∘g 的值域是把 f 作用于 g 的值域，但受到实际定义域的限制。务必检查排除值，例如分母为零或负数开平方。

4. Inverse Functions: Definition and Test | 反函数：定义与判定

A function f has an inverse f⁻¹ only if it is one-to-one on its given domain. Graphically, a one-to-one function passes the horizontal line test: every horizontal line cuts the graph at most once. The inverse function reverses the effect of f, so f⁻¹(f(x)) = x for all x in the domain of f.

函数 f 只有在给定定义域上是一一对应时才有反函数 f⁻¹。从图像上看，一一对应函数通过水平线检验：每一条水平线最多与图像相交一次。反函数逆转 f 的作用，因此对于 f 定义域内的所有 x，都有 f⁻¹(f(x)) = x。

5. Finding Inverse Functions Algebraically | 求反函数的代数方法

To find f⁻¹(x), start with y = f(x). Swap x and y to obtain x = f(y), then solve for y. For example, if f(x) = 2x – 3, write y = 2x – 3, swap to x = 2y – 3, and solve to get y = (x + 3)/2. Therefore f⁻¹(x) = (x + 3)/2. Always state the domain of f⁻¹ if the original domain is restricted.

求 f⁻¹(x) 时，先设 y = f(x)。交换 x 和 y 得到 x = f(y)，然后解出 y。例如，若 f(x) = 2x – 3，写 y = 2x – 3，交换为 x = 2y – 3，解得 y = (x + 3)/2。因此 f⁻¹(x) = (x + 3)/2。如果原定义域受限，务必说明 f⁻¹ 的定义域。

6. Graphs of Inverse Functions | 反函数的图像

The graph of y = f⁻¹(x) is a reflection of the graph of y = f(x) in the line y = x. Points swap coordinates: if (a, b) lies on y = f(x), then (b, a) lies on y = f⁻¹(x). The domain and range of f and f⁻¹ also swap. This symmetry is useful for sketching and checking algebraic work.

y = f⁻¹(x) 的图像是 y = f(x) 的图像关于直线 y = x 的反射。点的坐标互换：如果 (a, b) 在 y = f(x) 上，那么 (b, a) 在 y = f⁻¹(x) 上。f 和 f⁻¹ 的定义域与值域也互换。这种对称性对画图和检查代数运算都很有用。

7. Composite and Inverse Combined | 复合与反函数综合

When a function and its inverse are composed, they cancel each other out, provided the input is in the appropriate domain. For one-to-one f, both f(f⁻¹(x)) = x and f⁻¹(f(x)) = x hold. In exam questions, you may need to prove these identities or evaluate a composite such as f⁻¹(g⁻¹(x)), which equals (g∘f)⁻¹(x).

当一个函数与它的反函数复合时，只要输入在合适的定义域内，它们会相互抵消。对于一一对应的 f，f(f⁻¹(x)) = x 和 f⁻¹(f(x)) = x 都成立。在考试题中，你可能需要证明这些恒等式，或计算复合函数如 f⁻¹(g⁻¹(x))，它等于 (g∘f)⁻¹(x)。

8. Graph Transformations: Translation | 图像变换：平移

A translation shifts a graph without changing its shape. The transformation y = f(x) + a moves the graph up by a units if a > 0 and down by a units if a < 0. The transformation y = f(x + a) moves the graph left by a units if a > 0 and right by a units if a < 0. Notice the horizontal shift works in the opposite direction to the sign inside the bracket.

平移只移动图像而不改变形状。变换 y = f(x) + a 中，如果 a > 0，图像上移 a 个单位；如果 a < 0，图像下移 a 个单位。变换 y = f(x + a) 中，如果 a > 0，图像左移 a 个单位；如果 a < 0，图像右移 a 个单位。注意水平平移与括号内符号的方向相反。

9. Graph Transformations: Stretches and Reflections | 图像变换：伸缩与对称

Stretches and reflections change the scale or orientation of a graph. The following table summarises the most common transformations for Edexcel exams. In each case, c is a positive constant and c ≠ 0.

伸缩和对称改变图像的比例或方向。下表总结了 Edexcel 考试中最常见的变换。在每种情况中，c 为正数且 c ≠ 0。

y = c f(x): vertical stretch by scale factor c
y = f(cx): horizontal stretch by scale factor 1/c
y = -f(x): reflection in the x-axis
y = f(-x): reflection in the y-axis

对应的中文解释如下：y = c f(x) 表示沿竖直方向拉伸 c 倍；y = f(cx) 表示沿水平方向拉伸 1/c 倍；y = -f(x) 表示关于 x 轴对称；y = f(-x) 表示关于 y 轴对称。

10. Combining Transformations | 组合变换

When several transformations are applied to y = f(x), the order can affect the final result. For example, y = 2 f(x – 1) + 3 can be built in stages: start with y = f(x), translate right by 1 unit, stretch vertically by factor 2, then translate up by 3 units. Always describe transformations in the order that matches the algebraic structure of the expression.

当多个变换作用于 y = f(x) 时，顺序会影响最终结果。例如，y = 2 f(x – 1) + 3 可以分阶段构建：从 y = f(x) 开始，先向右平移 1 个单位，再沿竖直方向拉伸 2 倍，最后向上平移 3 个单位。描述变换时，始终按照表达式的代数结构顺序进行。

11. Common Edexcel Exam Pitfalls | 常见考试失分点

Students often lose marks by ignoring the domain of g when forming f(g(x)), by confusing f∘g with g∘f, or by forgetting that horizontal transformations move in the opposite direction. Another common error is writing f⁻¹(x) without swapping x and y properly, or not restricting the inverse for a non-one-to-one function. Always show your working and check with a point on the graph.

学生常见失分点包括：在构造 f(g(x)) 时忽略 g 的定义域，混淆 f∘g 与 g∘f，或忘记水平变换方向相反。另一个常见错误是没有正确交换 x 和 y 就写 f⁻¹(x)，或对非一一对应函数不限制反函数的定义域。务必写出步骤，并用图上的一个点进行检验。

12. Worked Example and Summary | 例题与总结

Worked example: Let f(x) = 2x – 3 and g(x) = x². Find f∘g(x), g∘f(x), and f⁻¹(x). First, f(g(x)) = f(x²) = 2x² – 3. Second, g(f(x)) = g(2x – 3) = (2x – 3)² = 4x² – 12x + 9. Third, for f⁻¹, write y = 2x – 3, swap to x = 2y – 3, solve to get y = (x + 3)/2. These three results show that algebra, order, and inverse steps must be handled systematically.

例题：设 f(x) = 2x – 3，g(x) = x²。求 f∘g(x)、g∘f(x) 和 f⁻¹(x)。首先，f(g(x)) = f(x²) = 2x² – 3。其次，g(f(x)) = g(2x – 3) = (2x – 3)² = 4x² – 12x + 9。第三，求 f⁻¹ 时，写 y = 2x – 3，交换为 x = 2y – 3，解得 y = (x + 3)/2。这三个结果表明代数运算、复合顺序和反函数步骤都必须系统地处理。

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Composite Functions and Graph Transformations | 复合函数与图像变换