Precalculus: Core Concepts and Study Guide | 预备微积分：核心知识点与学习指导

📚 Precalculus: Core Concepts and Study Guide | 预备微积分：核心知识点与学习指导

Precalculus is the essential course that consolidates algebra, geometry and trigonometry, preparing students for the rigorous analytical thinking required in calculus. It deepens understanding of functions, equations, and mathematical reasoning, laying a solid groundwork for advanced topics like limits, derivatives, and integrals.

预备微积分是巩固代数、几何和三角学的重要课程，帮助学生为微积分所需的严格分析思维做好准备。它加深对函数、方程和数学推理的理解，为极限、导数和积分等高级主题奠定坚实基础。

1. Functions and Transformations | 函数与变换

A function f maps each element x of the domain to exactly one element f(x) in the range. The vertical line test helps determine if a graph represents a function: any vertical line must intersect the graph at most once.

函数 f 将定义域中的每个元素 x 映射到值域中唯一的 f(x)。垂直线检验可用于判断图形是否表示函数：任何垂直线与图形最多相交一次。

Transformation types: f(x) + k (vertical shift), f(x – h) (horizontal shift), -f(x) (reflection across x-axis), f(-x) (reflection across y-axis). Stretching: a f(x) for vertical stretch/compression; f(bx) for horizontal stretch/compression.

变换类型：f(x) + k 垂直平移，f(x – h) 水平平移，-f(x) 关于 x 轴对称，f(-x) 关于 y 轴对称。伸缩变换：a f(x) 垂直伸缩；f(bx) 水平伸缩。

Composition of functions (f ∘ g)(x) = f(g(x)) requires that the range of g is within the domain of f. Inverse functions reflect across the line y = x and satisfy f(f^-1(x)) = x.

函数复合 (f ∘ g)(x) = f(g(x)) 要求 g 的值域在 f 的定义域内。反函数关于直线

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