📚 Review Set 4C: Mastering Functions and Transformations | 复习题集4C：精通函数与变换
Review Set 4C is a comprehensive collection of problems designed to solidify your understanding of functions, their graphs, and transformations. From basic domain and range to advanced logarithmic equations, this set covers the core skills needed for success in IB Mathematics. In this article, we will unpack the key concepts and common pitfalls encountered in Review Set 4C, providing clear explanations and worked examples.
复习题集4C是一套综合性练习题，旨在巩固你对函数、图像及变换的理解。从基础的定义域与值域到复杂的对数方程，这套题涵盖了IB数学成功所需的核心技能。本文将剖析复习题集4C中的关键概念和常见误区，提供清晰的解释和范例。
1. Functions and Domain/Range | 函数与定义域/值域
A function maps each input x to exactly one output f(x). When tackling Review Set 4C, you must be able to determine the domain (all possible x-values) and range (all possible y-values) from a given graph or algebraic expression. For rational functions, remember to exclude x-values that make the denominator zero; for square root functions, ensure the radicand is non-negative.
函数将每个输入x映射到唯一的输出f(x)。在处理复习题集4C时，你必须能从给定的图像或代数表达式中确定定义域（所有可能的x值）和值域（所有可能的y值）。对于有理函数，记得排除使分母为零的x值；对于平方根函数，要确保被开方数非负。
problems, you will meet functions like f(x) = 1/(x-2) where the domain is x ≠ 2, or g(x) = √x+3 domain in set notation or interval notation. For piecewise functions, check each piece separately and combine the restrictions.
在许多IB试题中，你会遇到如f(x)=1/(x-2)这样定义域为x≠2的函数，或g(x)=√x+3且定义域为x≥-3的函数。始终用集合记号或区间记号表示定义域。对于分段函数，需分别检查每一段并合并限制条件。
Example: For f(x) = √4 - x², the radicand 4 - x² ≥ 0 ⇒ x² ≤ 4 ⇒ root, so f(x) ≥ 0, and the maximum occurs when x=0, giving f(0)=2, so the range is [0, 2].
示例：对于f(x) = √4 - x²，被开方数4 - x² ≥ 0 ⇒ x² ≤ 4 ⇒ -2 ≤ x ≤ 2，因此定义域为[-2, 2]。输出为平方根，故f(x) ≥ 0，且当x=0时取得最大值f(0)=2，所以值域为[0, 2]。
2. Composite Functions | 复合函数
Composite functions involve applying one function to the result of another, denoted as (f∘ sks you to form composites, evaluate them at a given x, and state their domains. The domain of f∘g is the set of x in the domain of g such that g(x) is in the domain of f.
复合函数涉及将一个函数应用于另一个函数的结果，记为(f∘g)(x) = f(g(x))。复习题集4C经常要求构造复合函数，在给定x处求值，并给出其定义域。f∘g的定义域是g定义域中使得g(x)属于f定义域的x的集合。
Be careful: the order matters. (f∘g)(x) is generally not the same as (g∘f)(x). A classic mistake is to ignore the inner function's domain. Always start by finding the domain of the inside function g, then apply any restrictions imposed by f.
注意：顺序很关键。(f∘g)(x)通常不等于(g∘f)(x)。一个经典错误是忽略内层函数的定义域。始终先求内层函数g的定义域，再施加f带来的任何限制。
Example: If f(x) = 2x+1 and g(x) = √x, then (f∘g)(x) = f(√x) = 2√x+1, with domain x ≥ 0 because √x requires it. However, (g∘f)(x) = g(2x+1) = √2x+1, so the domain is 2x+1 ≥ 0 ⇒ x ≥ -1/2. Notice the difference.
示例：若f(x) = 2x+1，g(x) = √x，则(f∘g)(
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