IB Maths: Algebra and Functions Revision | IB 数学：代数和函数考点精讲

📚 IB Maths: Algebra and Functions Revision | IB 数学：代数和函数考点精讲

Algebra and functions form the backbone of the IB Mathematics curriculum, appearing in both Analysis & Approaches (AA) and Applications & Interpretation (AI). This guide consolidates the essential concepts, notation, and problem-solving techniques you need to master for exams, including domain and range, function transformations, polynomials, exponentials, and equation solving.

代数与函数是 IB 数学课程的核心，同时出现在分析与方法（AA）以及应用与解释（AI）的考纲中。这篇考点精讲将梳理定义域与值域、函数图像变换、多项式、指数对数以及方程求解等关键概念与解题技巧，帮助你有条理地备考。

1. Domain and Range | 定义域与值域

The domain of a function is the set of all possible input values (x) for which the function is defined. The range is the set of all possible output values (y). For real-valued functions, restrict the domain to avoid division by zero, negative values under even roots, or logarithms of non-positive numbers.

函数的定义域是所有可使函数有意义的输入值 x 的集合，值域是所有可能的输出值 y 的集合。对于实函数，需要限定定义域以避免分母为零、偶次根号下出现负数或对数的真数非正。

Example: For f(x) = √(x – 3) / (x – 5), the domain requires x – 3 ≥ 0 and x – 5 ≠ 0, giving x ≥ 3, x ≠ 5. The range can be found by considering the behaviour near the asymptote and the square root. Use interval notation: [3,5) ∪ (5,∞).

示例：对于 f(x) = √(x – 3) / (x – 5)，定义域要求 x – 3 ≥ 0 且 x – 5 ≠ 0，即 x ≥ 3，x ≠ 5。值域可通过分析渐近线附近的趋势和根式的取值范围得到。用区间表示为：[3,5) ∪ (5, ∞)。

2. Function Notation and Basics | 函数符号与基础概念

A function f maps each element x from the domain to exactly one element f(x) in the range. The vertical line test checks if a graph represents a function. Key vocabulary includes: one-to-one, many-to-one, and onto functions. The set of ordered pairs (x, f(x)) defines the function.

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)。竖线检验可判断一个图像是否表示函数。关键术语包含：一一函数、多对一函数和映成函数。有序数对 (x, f(x)) 的集合定义了该函数。

Function notation: f(x) = ax² + bx + c. To evaluate f(2), substitute x = 2. IB questions often involve piecewise functions where different rules apply over different intervals of the domain. Carefully identify which rule to use for a given input.

函数符号：f(x) = ax² + bx + c。计算 f(2) 时直接代入 x = 2。IB 试题常出现分段函数，在不同定义域区间上使用不同的表达式，必须仔细判断输入值对应的规则。

3. Composite Functions | 复合函数

The composite function (f ∘ g)(x) means first applying g, then applying f to the result: (f ∘ g)(x) = f(g(x)). The domain of the composite consists of all x in the domain of g such that g(x) is in the domain of f. Always find the inner function’s output first, then apply the outer function.

复合函数 (f ∘ g)(x) 表示先作用 g，再将结果代入 f：即 f(g(x))。复合函数的定义域包含所有属于 g 的定义域且使得 g(x) 落在 f 的定义域内的 x。务必先计算内层函数，再代入外层函数。

Example: Given f(x) = 2x + 1, g(x) = √(x), then (f ∘ g)(x) = 2√(x) + 1 with domain x ≥ 0. Note that (g ∘ f)(x) = √(2x + 1) has domain 2x + 1 ≥ 0 → x ≥ -1/2. The order matters greatly.

示例：已知 f(x) = 2x + 1，g(x) = √(x)，则 (f ∘ g)(x) = 2√(x) + 1，定义域 x ≥ 0。而 (g ∘ f)(x) = √(2x + 1) 的定义域为 2x + 1 ≥ 0 即 x ≥ -1/2。顺序至关重要。

4. Inverse Functions | 反函数

The inverse function f⁻¹(x) reverses the operation of f. To find the inverse, write y = f(x), swap x and y, then solve for y. The graph of f⁻¹ is the reflection of f in the line y = x. A function must be one-to-one to have an inverse over its entire domain; otherwise, restrict the domain.

反函数 f⁻¹(x) 逆转 f 的映射。求反函数的步骤：令 y = f(x)，交换 x 与 y，再解出 y。f⁻¹ 的图像是 f 关于直线 y = x 的反射。只有当函数为一一映射时，才在整个定义域上存在反函数；否则需限制定义域。

The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f. For f(x) = (x – 2)² + 1, x ≥ 2, the inverse is f⁻¹(x) = √(x – 1) + 2, with domain x ≥ 1. Always verify that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x within the restricted domains.

f⁻¹ 的定义域是 f 的值域，f⁻¹ 的值域是 f 的定义域。例如 f(x) = (x – 2)² + 1，x ≥ 2，其反函数为 f⁻¹(x) = √(x – 1) + 2，定义域 x ≥ 1。务必在限制域上验证 f(f⁻¹(x)) = x 及 f⁻¹(f(x)) = x。

5. Transformations of Graphs | 图像变换

Transformations allow you to sketch new functions from a parent graph. The main types are:

Translation: f(x) + k shifts vertically; f(x – h) shifts horizontally.
Stretch: a·f(x) stretches vertically by factor a; f(bx) compresses horizontally by factor 1/|b|.
Reflection: -f(x) reflects in the x-axis; f(-x) reflects in the y-axis.

Always apply horizontal transformations first (inside the bracket), then vertical transformations. Use mapping notation: (x, y) → (x/b + h, a·y + k).

图像变换让你能从基本初等函数图像快速画出新函数。主要类型包括：

平移：f(x) + k 垂直平移；f(x – h) 水平平移。
伸缩：a·f(x) 垂直伸缩 a 倍；f(bx) 水平压缩 1/|b| 倍。
反射：-f(x) 关于 x 轴对称；f(-x) 关于 y 轴对称。

通常先进行水平变换（括号内），再进行垂直变换。可使用映射记法：(x, y) → (x/b + h, a·y + k)。

Example: The graph of y = -2f(3(x + 1)) – 4 applied to f(x) = x² involves a horizontal compression by 1/3, translation left 1, vertical stretch by 2, reflection in x-axis, and translation down 4. Sketch step-by-step.

示例：对 f(x) = x² 应用变换 y = -2f(3(x + 1)) – 4，步骤为：水平压缩至 1/3，左移 1，垂直拉伸 2 倍，关于 x 轴反射，再下移 4。逐步画图。

6. Quadratic Functions | 二次函数

The standard form is f(x) = ax² + bx + c, a ≠ 0. The vertex form f(x) = a(x – h)² + k gives the vertex (h, k) and the axis of symmetry x = h. The discriminant Δ = b² – 4ac determines the nature of the roots: Δ > 0 two distinct real roots, Δ = 0 one repeated real root, Δ < 0 no real roots.

二次函数标准形式为 f(x) = ax² + bx + c，a ≠ 0。顶点式 f(x) = a(x – h)² + k 直接给出顶点 (h, k) 和对称轴 x = h。判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 有两个不同实根，Δ = 0 有两个相等实根，Δ < 0 无实根。

Completing the square converts standard form to vertex form: ax² + bx + c = a(x + b/(2a))² + (c – b²/(4a)). Use this to find the vertex, sketch the parabola, and solve quadratic equations. The quadratic formula x = (-b ± √Δ)/(2a) solves ax² + bx + c = 0.

配方法可将标准式化为顶点式：ax² + bx + c = a(x + b/(2a))² + (c – b²/(4a))。据此确定顶点、绘制抛物线以及解二次方程。求根公式 x = (-b ± √Δ)/(2a) 用于解 ax² + bx + c = 0。

7. Polynomials and the Factor Theorem | 多项式与因式定理

A polynomial of degree n takes the form P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₀. The Remainder Theorem: when P(x) is divided by (x – c), the remainder is P(c). The Factor Theorem: (x – c) is a factor of P(x) if and only if P(c) = 0. These are powerful tools for factorising cubics and higher degrees.

n 次多项式可表示为 P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₀。余数定理：P(x) 除以 (x – c) 的余数为 P(c)。因式定理：(x – c) 是 P(x) 的因式当且仅当 P(c) = 0。这两个定理是分解三次及以上多项式的利器。

To sketch a polynomial, find the y-intercept (set x = 0), factorise to find the x-intercepts (roots), and analyse the end behaviour determined by the leading term aₙxⁿ. A turning point exists where the derivative (for calculus-based questions) or by symmetry. Multiplicity of a root affects whether the graph crosses or touches the x-axis.

绘制多项式图像时，需要找出 y 轴截距（令 x = 0），通过因式分解求 x 轴截距（根），并根据首项 aₙxⁿ 分析趋于无穷的趋势。可通过求导（针对微积分题目）或利用对称性找拐点。根的重数决定图像与 x 轴相交还是相切。

8. Rational Functions | 有理函数

A rational function is a ratio of two polynomials: R(x) = P(x)/Q(x). Key features include vertical asymptotes where Q(x) = 0 (provided the factor does not cancel), horizontal or oblique asymptotes determined by the degrees of P and Q, and x- and y-intercepts. Always simplify the function first to remove common factors (holes).

有理函数是两个多项式的商：R(x) = P(x)/Q(x)。关键特征包括：Q(x) = 0 且不可约去时产生垂直渐近线；根据 P 与 Q 的次数关系确定水平或斜渐近线；以及 x 轴与 y 轴截距。务必先约去公因式以识别可去间断点（洞）。

For horizontal asymptotes: if degree(P) < degree(Q), y = 0; if degrees are equal, y = leading coeff. of P / leading coeff. of Q; if degree(P) > degree(Q), there is an oblique asymptote found by polynomial division. Sketch by plotting intercepts, asymptotes as dashed lines, and testing regions.

水平渐近线规则：若 deg(P) < deg(Q)，y = 0；若次数相等，y = P 的首项系数 / Q 的首项系数；若 deg(P) > deg(Q)，用多项式除法求出斜渐近线。作图时先标出截距，画出虚线渐近线，再分区检验符号。

9. Exponentials and Logarithms | 指数与对数

Exponential functions f(x) = aˣ (a > 0, a ≠ 1) have domain ℝ, range (0, ∞), and horizontal asymptote y = 0. The natural exponential eˣ is fundamental in growth and decay. Logarithmic functions g(x) = logₐ(x) are the inverses of exponentials, with domain (0, ∞), range ℝ, and vertical asymptote x = 0.

指数函数 f(x) = aˣ（a > 0，a ≠ 1）的定义域为 ℝ，值域 (0, ∞)，有水平渐近线 y = 0。自然指数 eˣ 是描述增长与衰减的基础。对数函数 g(x) = logₐ(x) 是指数函数的反函数，定义域 (0, ∞)，值域 ℝ，有垂直渐近线 x = 0。

Laws of logarithms: logₐ(xy) = logₐ(x) + logₐ(y), logₐ(x/y) = logₐ(x) – logₐ(y), logₐ(xⁿ) = n·logₐ(x). Change of base: logₐ(b) = log_c(b) / log_c(a). Use these to solve exponential equations, e.g., 2ˣ = 5 → x = log₂(5) = ln5 / ln2.

对数运算律：logₐ(xy) = logₐ(x) + logₐ(y)，logₐ(x/y) = logₐ(x) – logₐ(y)，logₐ(xⁿ) = n·logₐ(x)。换底公式：logₐ(b) = log_c(b) / log_c(a)。利用这些解指数方程，如 2ˣ = 5 → x = log₂(5) = ln5 / ln2。

10. Solving Equations and Inequalities | 解方程与不等式

For equations involving rational, radical, or exponential expressions, isolate the variable using inverse operations, always checking for extraneous solutions when squaring both sides or using other non-reversible steps. For inequalities, solve the corresponding equation first, then test intervals on a sign chart. Remember that multiplying or dividing by a negative number reverses the inequality sign.

对于含分式、根式或指数的方程，利用逆运算分离变量。若过程中两边平方或经过不可逆操作，务必检验增根。解不等式时，先解对应方程，再利用符号表检验各个区间。注意：乘或除以负数时，不等号方向要逆转。

Systems of equations: linear-quadratic systems often arise. Substitute one equation into the other to form a quadratic in one variable. The number of solutions corresponds to intersection points: 0, 1 (tangent), or 2 (secant). For exponential/logarithmic equations, use log laws to bring down exponents and check domain restrictions.

方程组：常出现一次与二次联立。将一次方程代入二次，化为一元二次方程求解。解的个数对应图像交点：0 个（相离）、1 个（相切）或 2 个（相交）。对于指数/对数方程，灵活运用对数法则降幂，并检查定义域限制。

Equation Type	Strategy	Example
Linear	Isolate x	3x – 7 = 2 → x = 3
Quadratic	Factor / formula	x² – 5x + 6 = 0 → x = 2, 3
Rational	Multiply by LCD; check extraneous	1/x + 1/(x-1) = 2 → x = 1 ± √2/2 (check x≠0,1)
Radical	Isolate radical, square both sides	√(x+2) = x – 4 → x = 7 (reject x=2)
Exponential	Take log both sides	3·2ˣ = 5 → 2ˣ = 5/3 → x = log₂(5/3)

上表总结了几类方程的解法与示例。

For inequalities like (x – 2)(x + 1) > 0, draw a number line, mark roots -1 and 2, test intervals to find x < -1 or x > 2. Always express the solution in interval notation.

对于如 (x – 2)(x + 1) > 0 的不等式，画数轴，标注根 -1 和 2，测试区间得到 x < -1 或 x > 2。务必用区间表示解集。

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IB Maths: Algebra and Functions Revision | IB 数学：代数和函数 考点精讲