📚 Exercise B.3: Composite and Inverse Functions | 练习B.3：复合函数与反函数
Composite and inverse functions are central to the IB Mathematics curriculum, appearing regularly in both Analysis and Approaches and Applications and Interpretation. Exercise B.3 typically challenges you to build fluency in combining functions, untangling their domains, and reversing the input‑output mapping. This article unpacks every key idea, common mistake, and typical exam‑style question so that you can approach the exercise with confidence.
复合函数与反函数是IB数学课程的核心内容，频繁出现在分析与方法和应用与解释两种方向中。练习B.3通常要求你熟练组合函数、理清定义域，并逆转输入与输出的对应关系。本文将逐一拆解每个关键概念、常见错误和典型考题，让你能够自信地完成这个练习。
1. What Is a Composite Function? | 什么是复合函数？
A composite function is created when one function is applied to the result of another function. If we have two functions f(x) and g(x), the composite function f(g(x)) means we first apply g to x, then apply f to the output of g. Visually, think of a pipeline: x enters g, the result g(x) exits, then that result immediately enters f, producing f(g(x)).
当我们将一个函数作用在另一个函数的结果上时，就产生了复合函数。如果有两个函数 f(x) 和 g(x)，复合函数 f(g(x)) 表示先将 g 作用到 x 上，接着将 g 的输出作为 f 的输入。可以形象地想象成一条流水线：x 进入 g，结果 g(x) 流出，紧接着流入 f，最终得到 f(g(x))。
2. Notation and Calculation | 符号与计算
The notation (f ∘ g)(x) is read as 'f composed with g of x' f(x) = 2x + 3 and g(x) = x², then f(g(4)) = f(16) = 2(16) + 3 = 35. The expression for (f ∘ g)(x) is f(x²) = 2x² + 3. Notice that order matters: (g ∘ f)(x) = g(2x+3) = (2x+3)², which is different.
符号 (f ∘ g)(x) 读作 'f 复合 g 在 x 处的值'，与 f(g(x)) 完全等价。代入具体数值时，总是从内向外计算。例如，设 f(x) = 2x + 3，g(x) = x²，那么 f(g(4)) = f(16) = 2×16 + 3 = 35。函数的表达式为 (f ∘ g)(x) = f(x²) = 2x² + 3。注意顺序很重要：(g ∘ f)(x) = g(2x+3) = (2x+3)²，结果完全不同。
3. Domain Considerations | 定义域注意事项
The domain of a composite function f(g(x)) is the set of all x in the domain of g such that g(x) belongs to the domain of f. Always start by finding the domain of the inner function g, and then restrict it further so that the output from g is a valid input for f. A common IB question asks: 'Find the largest possible domain of f ∘ g.' You must exclude any x‑values that cause g to be undefined, or that make g(x) fall outside the domain of f.
复合函数 f(g(x)) 的定义域是所有属于 g 的定义域，且使得 g(x) 属于 f 的定义域的 x 的集合。解决这类问题时，先找出内层函数 g 的定义域，再进一步限制，使 g 的输出在 f 的定义域内。IB考试中常见的问题是：“求 f ∘ g 的最大可能定义域。”你必须排除导致 g 无定义的 x 值，或使得 g(x) 超出 f 定义域的 x 值。
4. Composite Functions Involving More Than Two Functions | 涉及三个及以上函数的复合
Composition can be extended to three or more functions, such as (h ∘ g ∘ f)(x) = h(g(f(x))). The evaluation remains inside‑ lgebraic simplification, work from the innermost function outward. Associativity holds: (h ∘ g) ∘ f = h ∘ (g ∘ f), so you can group sequentially with confidence.
复合可以扩展到三个或更多函数，例如 (h ∘ g ∘ f)(x) = h(g(f(x)))。求值仍然是从内向外：先作用 f，再对结果作用 g，最后作用 h。进行代数化简时，从最内层函数逐步向外处理。复合运算满足结合律：(h ∘ g) ∘ f = h ∘ (g ∘ f)，因此你可以放心地按顺序分组计算。
5. Real‑World Modelling with Composite Functions | 用复合函数进行现实建模
Composite functions often model situations where one process feeds into another. In IB exams, you might see a problem about converting currencies, applying a tax rate after a discount, or calculating the volume of a shape after a transformation. For instance, if g(t) gives the radius of a balloon after t seconds, and f(r) gives the volume from the radius, then f(g(t)) directly models volume change over time.
复合函数常用于描述一个过程的结果进入下一个过程的实际情景。IB考试中可能会出现货币兑换、先打折后加税，或经变换后求图形体积的问题。例如，g(t) 给出气球经过 t 秒后的半径，f(r) 给出由半径得到的体积，那么 f(g(t)) 就直接建立了体积随时间变化的模型。
6. Introduction to Inverse Functions | 反函数介绍
An inverse function, denoted f⁻¹(x), reverses the effect of the original function. If f maps a to b, then f⁻¹ maps b back to a. For a function to have an inverse that is also a function, it must be one‑to‑one (each output comes from exactly one input). The horizontal line test on the graph of f confirms this: if any horizontal line cuts the graph more than once, f is not one‑to‑one and does not have an inverse function without restricting the domain.
反函数记作 f⁻¹(x)，它逆转原函数的作用。如果 f 将 a 映射到 b，那么 f⁻¹ 就将 b 映射回 a。一个函数要有反函数，必须是一一对应的（每个输出都恰好对应一个输入）。在 f 的图像上使用水平线检验可以确认这一点：如果存在一条水平线与图像相交多于一点，那么 f 就不是一一对应函数，除非限制定义域，否则不存在反函数。
7. Finding the Inverse Function Algebraically | 通过代数方法求反函数
− 7, swap to x = 3y − 7, solve for y: 3y = x + 7 → y = (x + 7)/3. Thus f⁻¹(x) = (x + 7)/3.
求 y = f(x) 反函数的标准方法是：写出 y = f(x)，交换 x 和 y 得到 x = f(y)，然后解出用 x 表示的 y。这个结果就是 y = f⁻¹(x)。例如，求 f(x) = 3x − 7 的反函数：令 y = 3x − 7，交换得 x = 3y − 7，解 y：3y = x + 7 → y = (x + 7)/3。因此 f⁻¹(x) = (x + 7)/3。
8. Domain and Range of Inverse Functions | 反函数的定义域与值域
A crucial relationship holds: the domain of f⁻¹ is exactly the range of f, and the range of f⁻¹ is exactly the domain of f. This is because the inverse function simply swaps inputs and outputs. When solving an IB problem, always state the domain of the inverse carefully – it is not always all real numbers. You must consider any restrictions originally present in the range of f.
存在一个至关重要的关系：f⁻¹ 的定义域正是 f 的值域，而 f⁻¹ 的值域正是 f 的定义域。这是因为反函数本质上交换了输入和输出。在解答IB问题时，一定要仔细说明反函数的定义域——它未必是全体实数。你必须考虑原函数值域中已有的任何限制。
9. Graphs of a Function and Its Inverse | 函数与其反函数的图像
h of f corresponds to a point (b, a) on the graph of f⁻¹. If a function is its own inverse – for example, f(x) = 1/x or f(x) = −x – uestions frequently ask you to sketch both graphs on the same axes and label key features.
f⁻¹ 的图像是 f 的图像关于直线 y = x 的反射。f 图像上的每一点 (a, b) 都对应 f⁻¹ 图像上的一点 (b, a)。如果一个函数是它自身的反函数，例如 f(x) = 1/x 或 f(x) = −x，其图像将关于直线 y = x 对称。IB考题经常要求你在同一坐标系中画出两个函数的图像并标注关键特征。
10. Verifying Inverse Functions | 验证反函数
the domain of f. If both hold, then g = f⁻¹. This test is particularly useful in multiple‑choice sections where you can quickly substitute expressions.
要验证两个函数 f 和 g 是否互为反函数，需要检验两个复合恒等式：对所有 g 定义域内的 x，有 f(g(x)) = x；并且对所有 f 定义域内的 x，有 g(f(x)) = x。如果两者都成立，那么 g = f⁻¹。这种检验在选择题部分尤其有用，因为你可以快速代入表达式进行验证。
11. Common Pitfalls and Tips for Exercise B.3 | 练习B.3的常见错误与建议
composite or inverse function. When solving for an inverse, remember that the inverse must pass the vertical line test – lation produces a ± square root, you must restrict the sign based on the original function' en working with trigonometric functions, restrict domains appropriately so inverses exist.
常见错误之一是忘记 f(g(x)) 与 g(f(x)) 并不相同。同学们在写出复合或反函数的最终答案时，也常常忽略定义域。求反函数时请记住，反函数必须通过垂直线检验——如果代数运算出现了平方根的±符号，必须根据原函数的值域限定符号。务必清晰标注最终答案：f⁻¹(x) = ...，在处理三角函数时，要适当限制定义域以使反函数存在。
12. Conclusion | 小结
Mastering Exercise B.3 requires a clear mental model of function pipelines and a systematic approach to domains. By practising the algebraic swap‑and‑solve technique, drawing connections between composite and inverse graphs, and always checking domains, you will be well prepared for any IB question. The key is to treat functions as machines that can be chained and reversed, making sure each step is mathematically valid.
掌握练习B.3需要你对函数流水线有清晰的心理模型，并对定义域采取系统化的处理方法。通过练习代数上的交换和解方程技巧，建立复合与反函数图像间的关联，并时刻检查定义域，你将能为任何IB考题做好充分准备。关键在于把函数视为可串联并可逆转的机器，并确保每一步在数学上都是成立的。
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