Algebra and Functions Core Revision | 代数和函数考点精讲

📚 Algebra and Functions Core Revision | 代数和函数考点精讲

In the IB and WJEC mathematics syllabuses, Algebra and Functions form the backbone of many advanced topics. This article walks you through the key concepts you must master, from basic function notation and transformations to solving equations involving polynomials, exponentials, logarithms, and trigonometric functions. Each section pairs an English explanation with its Chinese equivalent to support bilingual revision.

在 IB 和 WJEC 数学课程中，代数和函数是许多高级专题的基石。本文带你梳理必须掌握的核心概念，从基本的函数记号和图像变换，到解多项式、指数、对数与三角方程，每个要点都采用英中对照讲解，方便双语复习。

1. Function Notation and Domain/Range | 函数记号与定义域、值域

A function maps each element from a set of inputs (the domain) to exactly one element in a set of outputs (the range). We often write f(x) = 2x + 3. The domain is the set of all allowable x-values, while the range is the set of all possible y-values produced by the function. For example, f(x) = √x has domain x ≥ 0 and range y ≥ 0.

函数将定义域（输入集）中的每一个元素恰好映射到值域（输出集）中的一个元素。我们常写作 f(x) = 2x + 3。定义域是所有容许的 x 值的集合，值域是由函数产生的所有可能 y 值的集合。例如，f(x) = √x 的定义域是 x ≥ 0，值域是 y ≥ 0。

2. Composite Functions | 复合函数

Composite functions combine two or more functions. If f(x) = x² and g(x) = 2x + 1, then the composite f(g(x)), also written as (f o g)(x), means first apply g, then apply f to the result: f(g(x)) = (2x + 1)². Always check the domain of the inner function and the resulting expression to avoid undefined values. Note that f(g(x)) is generally not equal to g(f(x)).

复合函数将两个或多个函数组合起来。若 f(x) = x², g(x) = 2x + 1，则复合函数 f(g(x))，也写作 (f o g)(x)，表示先作用 g 再将结果作用于 f：f(g(x)) = (2x + 1)²。务必检查内层函数及最终表达式的定义域，避免出现无定义的值。注意 f(g(x)) 通常不等于 g(f(x))。

3. Inverse Functions | 反函数

An inverse function f⁻¹ reverses the effect of f. For a function to have an inverse, it must be one-to-one (pass the horizontal line test). To find the inverse, swap x and y in the equation y = f(x) and then solve for y. For instance, if f(x) = (x – 1)/2, swap to x = (y – 1)/2 and solve to get f⁻¹(x) = 2x + 1. The graphs of f and f⁻¹ are reflections in the line y = x. The domain of f⁻¹ is the range of f, and vice versa.

反函数 f⁻¹ 逆转 f 的作用。只有一一对应的函数（通过水平线检验）才存在反函数。求反函数的方法是将 y = f(x) 中的 x 和 y 互换，再解出 y。例如，若 f(x) = (x – 1)/2，互换得到 x = (y – 1)/2，解得 f⁻¹(x) = 2x + 1。f 和 f⁻¹ 的图像关于直线 y = x 对称。f⁻¹ 的定义域正是 f 的值域，反之亦然。

4. Graph Transformations | 图像变换

Function graphs can be transformed by simple operations. f(x) + a shifts the graph vertically by a units. f(x + a) shifts it horizontally by -a units (left if a > 0). af(x) stretches it vertically by factor a; f(ax) stretches it horizontally by factor 1/a. Negative signs cause reflections: -f(x) reflects in the x-axis, f(-x) reflects in the y-axis. Mastering these patterns helps you sketch curves quickly without plotting many points.

函数图像可以通过简单运算进行变换。f(x) + a 使图像沿竖直方向平移 a 个单位。f(x + a) 沿水平方向平移 -a 个单位（a > 0 时向左）。af(x) 使图像竖直方向伸缩，伸缩因子为 a；f(ax) 则水平方向伸缩，伸缩因子为 1/a。负号产生对称：-f(x) 关于 x 轴对称，f(-x) 关于 y 轴对称。掌握这些规律后，无需描点即可快速绘制曲线。

5. Polynomial Functions | 多项式函数

Polynomial functions are expressions of the form aₙxⁿ + … + a₁x + a₀. Key topics include factorising, the Factor Theorem, and the Remainder Theorem. For a polynomial p(x), if p(a) = 0 then (x – a) is a factor. The Remainder Theorem states that when p(x) is divided by (x – a), the remainder is p(a). Graphing polynomials involves identifying x-intercepts (roots), y-intercept, end behaviour, and turning points. The degree indicates the maximum number of real roots and the number of possible turning points.

多项式函数是形如 aₙxⁿ + … + a₁x + a₀ 的表达式。核心考点包括因式分解、因式定理和余式定理。对于多项式 p(x)，若 p(a) = 0，则 (x – a) 是一个因式。余式定理指出，当 p(x) 除以 (x – a) 时，余数为 p(a)。绘制多项式图像需要确定 x 轴截距（根）、y 轴截距、末端走势和极值点。次数决定了最多实根个数和可能的极值点个数。

6. Rational Functions and Asymptotes | 有理函数与渐近线

Rational functions are ratios of two polynomials, e.g. f(x) = (2x + 1)/(x – 3). Their graphs often feature vertical asymptotes where the denominator is zero (and the numerator non-zero), and horizontal or oblique asymptotes determined by the degrees of numerator and denominator. For equal degrees, the horizontal asymptote is y = (leading coefficient of numerator)/(leading coefficient of denominator). For rational functions, also identify any holes (removable discontinuities) where both numerator and denominator share a factor. Sketching asymptotes and intercepts provides a clear graph.

有理函数是两个多项式的比，例如 f(x) = (2x + 1)/(x – 3)。它们的图像在分母为零（且分子非零）处常有竖直渐近线，并根据分子分母的次数确定水平渐近线或斜渐近线。当次数相等时，水平渐近线为 y = (分子首项系数)/(分母首项系数)。还要识别可去间断点（洞），即分子分母有公因式的位置。绘出渐近线和截距即可得到清晰的图像。

7. Exponential and Logarithmic Functions | 指数函数与对数函数

Exponential functions of the form f(x) = aˣ (with a > 0, a ≠ 1) model growth and decay. The natural exponential function is f(x) = eˣ. The inverse function is the logarithm: logₐ(x). Key properties include logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx – logₐy, and logₐ(xⁿ) = n logₐx. The natural logarithm is ln x = logₑ x. Solving equations like e²ˣ = 5 requires taking logs: 2x = ln 5, thus x = (ln 5)/2.

指数函数形如 f(x) = aˣ（a > 0, a ≠ 1）用于描述增长与衰减。自然指数函数为 f(x) = eˣ。其反函数是对数函数：logₐ(x)。关键性质包括 logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx – logₐy, logₐ(xⁿ) = n logₐx。自然对数记为 ln x = logₑ x。解方程如 e²ˣ = 5 需要两边取对数：2x = ln 5，故 x = (ln 5)/2。

8. Trigonometric Functions and Identities | 三角函数与恒等式

The three primary trigonometric functions are sine (sin θ), cosine (cos θ), and tangent (tan θ). In the unit circle, sin θ and cos θ represent the y– and x-coordinates of a point on the circle. Key identities include sin²θ + cos²θ = 1, tan θ = sin θ / cos θ, and the double-angle formulas: sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ – sin²θ. Periodicity and symmetry (CAST diagram) help solve trig equations, such as finding all solutions in a given interval.

三个基本三角函数是正弦（sin θ）、余弦（cos θ）和正切（tan θ）。在单位圆中，sin θ 和 cos θ 分别表示圆上点的 y 坐标和 x 坐标。核心恒等式包括 sin²θ + cos²θ = 1、tan θ = sin θ / cos θ，以及倍角公式：sin 2θ = 2 sin θ cos θ、cos 2θ = cos²θ – sin²θ。周期性和对称性（CAST 图）有助于求解三角方程，例如在指定区间内找出所有解。

9. Solving Equations and Inequalities | 方程与不等式的求解

Algebraic techniques are essential for solving a wide range of equations. For quadratics, use factorising, completing the square, or the quadratic formula x = [-b ± √(b² – 4ac)] / (2a). Simultaneous equations can be solved by substitution or elimination. Inequalities require careful treatment: multiplying or dividing by a negative number reverses the inequality sign. For polynomial inequalities, sketch the graph or use a sign table. When dealing with rational inequalities, move all terms to one side and consider critical values where the expression is zero or undefined.

代数技巧在求解各类方程时至关重要。对于二次方程，可用因式分解、配方法或求根公式 x = [-b ± √(b² – 4ac)] / (2a)。联立方程组可通过代入法或消元法求解。处理不等式时要小心：乘或除一个负数会反转不等号方向。对于多项式不等式，可绘制图像或使用符号表。处理分式不等式时，把所有项移到一边，并考虑表达式为零或无定义的临界值。

10. Function Analysis through Calculus Connections | 结合微积分的函数分析

Though calculus may be a separate topic, understanding the link between algebra and derivatives helps analyse functions. The derivative f'(x) gives the gradient and identifies increasing/decreasing intervals, while the second derivative f”(x) indicates concavity and points of inflection. Stationary points occur where f'(x) = 0; use the second derivative test or a sign chart to classify them as maxima, minima, or points of inflection. Algebraic manipulation of derivative equations often returns students to the core skills of solving equations and simplifying expressions.

虽然微积分可能是独立专题，但理解代数与导数的联系有助于分析函数。导数 f'(x) 给出斜率并确定递增、递减区间，二阶导数 f”(x) 揭示凹凸性与拐点。驻点处 f'(x) = 0；利用二阶导数检验或符号表将其分为极大值点、极小值点或拐点。对导数方程进行代数运算时，往往需要用到解方程与化简表达式的核心技能。

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Algebra and Functions Core Revision | 代数和函数 考点精讲