📚 Topic 2: Functions – HL Only | 主题2：函数 – 仅限HL
In IB Mathematics: Analysis and Approaches HL, Topic 2 extends the study of functions beyond the Standard Level syllabus. This advanced material covers composite functions with careful domain analysis, inverse functions and one-to-one restrictions, self-inverse and odd/even functions, polynomial theorems, synthetic division, rational functions and their asymptotes, absolute value equations and inequalities, and more complex exponential and logarithmic equations. Mastery of these HL-only topics is essential for success in Paper 1 and Paper 2, and forms a foundation for calculus and further mathematics.
在IB数学分析与方法HL课程中，主题2在标准水平的基础上进一步深化函数的学习。这些高阶内容涵盖仔细分析定义域的复合函数、反函数与一一映射限制、自反函数、奇偶函数、多项式定理、综合除法、有理函数及其渐近线、绝对值方程与不等式，以及更复杂的指数与对数方程。掌握这些仅限HL的专题对于试卷一和试卷二的成功至关重要，并为微积分和高等数学奠定基础。
1. Composite Functions and Domain Restrictions | 复合函数与定义域限制
For two functions f and g, the composite function (f ∘ (x) is in the domain of f. It is crucial to determine the domain of the composite function by considering both the domain of the inner function g and the requirement that g(x) belongs to the domain of f. A common mistake is to only look at the final expression without checking these restrictions.
对于两个函数 f 和 g，复合函数 (f ∘ g)(x) = f(g(x)) 只有当 g(x) 属于 f 的定义域时才有定义。求复合函数的定义域时，必须同时考虑内层函数 g 的定义域，以及 g(x) 必须落在 f 的定义域内这一要求。常见的错误是只看最终的函数表达式，而忽略了这些限制条件。
Example: Let f(x) = √x with domain x ≥ 0, and g(x) = x − 5 with domain ℝ. Then (f ∘ g)(x) = √x − 5 ven though g is defined for all real numbers, the square root restricts the composite domain.
示例：设 f(x) = √x，定义域为 x ≥ 0；g(x) = x − 5，定义域为全体实数。则 (f ∘ g)(x) = √x − 5 的定义域要求 x − 5 ≥ 0，即 x ≥ 5。尽管 g 对所有实数都有定义，但平方根限制了复合函数的定义域。
2. Inverse Functions and One-to-One Restriction | 反函数与一一映射限制
l domain, we must restrict the domain to make it invertible. For example, f(x) = x² is not one-to-one over ℝ, but by restricting to x ≥ 0, the inverse f⁻¹(x) = √x exists.
函数 f 有反函数 f⁻¹ 当且仅当 f 是一一映射（单射）。求反函数时，交换 x 和 y 并解出 y。f⁻¹ 的定义域是 f 的值域，f⁻¹ 的值域是 f 的定义域。从图像上看，反函数是关于直线 y = x 的反射。如果函数在其自然定义域上不是一一映射，我们必须限制定义域使其可逆。例如，f(x) = x² 在全体实数上不是单射，但若限制定义域为 x ≥ 0，则存在反函数 f⁻¹(x) = √x。
olds for all x in the domain of f, and f(f⁻¹(x)) = x holds for all x in the domain of f⁻¹.
关系式 f⁻¹(f(x)) = x 对所有 f 定义域内的 x 成立，而 f(f⁻¹(x)) = x 对所有 f⁻¹ 定义域内的 x 成立。
3. Self-Inverse Functions | 自反函数
= (ax + b)/(cx − a) under certain conditions. These functions often appear in exam questions requiring verification and graph analysis.
若一个函数满足 f⁻¹(x) = f(x)，即它的反函数是它自身，则称其为自反函数。从图像上看，自反函数关于直线 y = x 对称。代数上，可以通过证明 f(f(x)) = x 来验证。常见的例子包括 f(x) = c − x、f(x) = k/x（需限制定义域），以及在某些条件下 f(x) = (ax + b)/(cx − a)。这类函数经常出现在考题中，要求验证并分析图像。
4. Odd and Even Functions | 奇函数与偶函数
ts domain. Odd functions are symmetric about the origin. Many functions are neither odd nor even. Understanding symmetry helps in graphing and integrating functions later. Polynomials with only even powers are even; those with only odd powers are odd. The product of two odd functions is even, and the product of an even and an odd function is odd.
函数 f 是偶函数，若对其定义域内所有 x 都有 f(−x) = f(x)。偶函数关于 y 轴对称。函数是奇函数，若对其定义域内所有 x 都有 f(−x) = −f(x)。奇函数关于原点对称。许多函数既非奇也非偶。理解对称性有助于函数的作图以及后续的积分。只有偶次幂的多项式是偶函数；只有奇次幂的多项式是奇函数。两个奇函数的乘积是偶函数，一个偶函数与一个奇函数的乘积是奇函数。
5. Factor Theorem and Remainder Theorem | 因式定理与余式定理
ems are essential for factorising cubics and higher-degree polynomials, and for solving polynomial equations. They allow us to test possible rational roots from the factors of
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