📚 Common Special Functions and Their Properties in IB Mathematics | IB数学：常见特殊函数及其性质
In IB Mathematics, students are expected to understand a range of special functions — their definitions, domains, ranges, graphs, symmetries, asymptotes, and transformations. This article summarises the most common special functions and their key properties for both Analysis and Approaches (AA) and Applications and Interpretation (AI) syllabi.
在IB数学中，学生需要掌握一系列常见特殊函数的定义、定义域、值域、图像、对称性、渐近线以及变换，这些内容同时覆盖分析与方法（AA）与应用与解释（AI）两大课程体系。本文将系统梳理这些函数的典型性质，帮助考生快速复习与备考。
1. Linear and Constant Functions | 线性函数与常数函数
A linear function has the form f(x) = ax + b, where a and b are constants. Its graph is a straight line with slope a and y-intercept b. The domain and range are both ℝ (all real numbers). It is one-to-one if a ≠ 0, and its inverse is also linear.
线性函数的一般形式为 f(x) = ax + b，其中 a 和 b 为常数。其图像是一条直线，斜率为 a，y 轴截距为 b。定义域和值域均为全体实数 ℝ。当 a ≠ 0 时，函数是单射，且其反函数仍为线性函数。
- Constant function: f(x) = c; range = {c}; graph is a horizontal line.
- 常数函数：f(x) = c；值域为 {c}；图像为水平直线。
- Zero slope: a = 0 gives constant; undefined slope not allowed for functions.
- 零斜率：a = 0 时为常数函数；函数不允许斜率为无穷大。
2. Quadratic Functions | 二次函数
A quadratic function has the form f(x) = ax² + bx + c (a ≠ 0). Its graph is a parabola. The vertex occurs at x = -b/(2a). The discriminant Δ = b² - 4ac determines the number of real roots: two, one, or none.
二次函数的一般形式为 f(x) = ax² + bx + c（a ≠ 0）。其图像为抛物线。顶点横坐标为 x = -b/(2a)。判别式 Δ = b² - 4ac 决定实根个数：两个、一个或无实根。
- If a > 0, the parabola opens upward and has a minimum; if a < 0, it opens downward and has a maximum.
- 若 a > 0，抛物线开口向上，存在最小值；若 a < 0，开口向下，存在最大值。
- The domain is ℝ; the range is [yvertex, ∞) or (-∞, yvertex].
- 定义域为 ℝ；值域为 [y_顶点, ∞) 或 (-∞, y_顶点]。
- Axis of symmetry: x = -b/(2a). It is an even function only if b = 0.
- 对称轴：x = -b/(2a)。只有当 b = 0 时才是偶函数。
3. Power and Polynomial Functions | 幂函数与多项式函数
A power function is f(x) = xⁿ for n ∈ ℝ. If n is a positive integer, it is a monomial. A polynomial function is a sum of monomials: P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where aₙ ≠ 0. The degree n determines the end behaviour.
幂函数形如 f(x) = xⁿ，其中 n ∈ ℝ。当 n 为正整数时即为单项式。多项式函数是若干单项式之和：P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀，其中 aₙ ≠ 0。次数 n 决定函数的端部行为。
- Even powers: f(x) = x² and x⁴ are even functions; odd powers are odd functions.
- 偶次幂：x² 与 x⁴ 为偶函数；奇次幂为奇函数。
- The fundamental theorem of algebra: a polynomial of degree n has exactly n complex roots.
- 代数基本定理：n 次多项式恰有 n 个复数根。
- End behaviour: as x → ±∞, the leading term dominates.
- 端部行为：当 x → ±∞ 时，首项主导函数变化。
4. Rational Functions | 有理函数
A rational function is f(x) = P(x)/Q(x) he factor cancels), horizontal or oblique asymptotes determined by the degrees of P and Q.
有理函数形如 f(x) = P(x)/Q(x)，其中 P 与 Q 为多项式且 Q(x) ≠ 0。关键特征包括：当 Q(x) = 0 时出现竖直渐近线（若该因子未约分），而水平或斜渐近线由 P 与 Q 的次数决定。
| Degree comparison | Asymptote |
| deg P < deg Q | y = 0 |
| deg P = deg Q | y = leading coefficient ratio |
| deg P = deg Q + 1 | Oblique (slant) asymptote |
- Domain excludes zeros of Q(x).
- 定义域排除 Q(x) 的零点。
- A hole occurs if a common factor cancels.
- 若存在公共因子相约，则形成可去间断点（洞）。
5. Radical Functions | 根式函数
Radical functions involve roots, typically f(x) = √x or more generally f(x) = (g(x))1/n. For even roots, the radicand must be non-negative; for odd roots, the domain can be all real numbers.
根式函数包含根号，常见的有 f(x) = √x，更一般地 f(x) = (g(x))1/n。对于偶次根，被开方数必须非负；对于奇次根，定义域可为全体实数。
- f(x) = √x has domain [0, ∞), range [0, ∞), and is increasing.
- f(x) = √x 的定义域为 [0, ∞)，值域为 [0, ∞)，且单调递增。
- The graph is a vertical reflection of y = x² for x ≥ 0.
- 其图像是 y = x²（x ≥ 0）关于直线 y = x 的反射。
- For cube root f(x) = x1/3, domain and range are both ℝ, and the function is odd.
- 对于立方根 f(x) = x1/3，定义域和值域均为 ℝ，且为奇函数。
6. Absolute Value Functions | 绝对值函数
The absolute value function is defined as |x| = x if x ≥ 0, and -x if x < 0. Its graph is V-shaped with a sharp corner at the origin. The transformation f(x) = a|x - h| + k shifts the vertex to (h, k).
绝对值函数定义为 |x| = x（当 x ≥ 0），-x（当 x < 0）。其图像为 V 形，在原点处有一个尖角。变换 f(x) = a|x - h| + k 将顶点移至 (h, k)。
- Domain: ℝ; range: [0, ∞) for the basic function.
- 定义域：ℝ；基本函数的值域为 [0, ∞)。
- It is an even function: |x| = |-x|.
- 它是偶函数：|x| = |-x|。
- It is not differentiable at x = 0.
- 它在 x = 0 处不可导。
7. Exponential Functions | 指数函数
An exponential function has the form f(x) = a·bˣ, where b > 0, b ≠ 1, and a ≠ 0. The base b determines growth (b > 1) or decay (0 < b < 1). The natural exponential is f(x) = eˣ.
指数函数形如 f(x) = a·bˣ，其中 b > 0，b ≠ 1，a ≠ 0。底数 b 决定增长（b > 1）或衰减（0 < b < 1）。自然指数函数为 f(x) = eˣ。
- Domain: ℝ; range: (0, ∞) for a > 0.
- 定义域：ℝ；当 a > 0 时值域为 (0, ∞)。
- Horizontal asymptote: y = 0.
- 水平渐近线：y = 0。
- The y-intercept is (0, a). There is no x-intercept.
- y 轴截距为 (0, a)，没有 x 轴截距。
- Laws of exponents apply: bˣ·bʸ = bˣ⁺ʸ, (bˣ)ʸ = bˣʸ.
- 指数法则适用：bˣ·bʸ = bˣ⁺ʸ，(bˣ)ʸ = bˣʸ。
8. Logarithmic Functions | 对数函数
The logarithmic function is the inverse of the exponential function: y = log_b(x) iff x = bʸ. The natural logarithm is ln x = log_e x. Its domain is (0, ∞) and range is ℝ.
对数函数是指数函数的反函数：y = log_b(x) 等价于 x = bʸ。自然对数为 ln x = log_e x。其定义域为 (0, ∞)，值域为 ℝ。
- Vertical asymptote: x = 0.
- 竖直渐近线：x = 0。
- The x-intercept is (1, 0).
- x 轴截距为 (1, 0)。
- It is increasing if b > 1, decreasing if 0 < b < 1.
- 当 b > 1 时单调递增；当 0 < b < 1 时单调递减。
- ies: log_b(xy) = log_b x + log_b y; log_b(x/y) = log_b x - log_b y; log_b(xⁿ) = n log_b x.
- 重要性质：log_b(xy) = log_b x + log_b y；log_b(x/y) = log_b x - log_b y；log_b(xⁿ) = n log_b x。
9. Trigonometric Functions | 三角函数
The basic trigonometric functions are sin x, cos x, tan x, together with reciprocal functions csc x, sec x, cot x. They are periodic and defined using the unit circle. Their graphs show amplitude, period, and phase shifts.
基本三角函数为 sin x、cos x、tan x，以及倒数函数 csc x、sec x、cot x。它们具有周期性，并借助单位圆定义。图像体现振幅、周期与相位移。
| Function | Domain | Range | Period |
| sin x | ℝ | [-1, 1] | 2π |
| cos x | ℝ | [-1, 1] | 2π |
| tan x | x ≠ π/2 + kπ | ℝ | π |
- sin x is odd; cos x is even; tan x is odd.
- sin x 为奇函数；cos x 为偶函数；tan x 为奇函数。
- tan x has vertical asymptotes at x = π/2 + kπ.
- tan x 在 x = π/2 + kπ 处有竖直渐近线。
- The Pythagorean identity: sin²x + cos²x = 1.
- 毕达哥拉斯恒等式：sin²x + cos²x = 1。
10. Piecewise Functions | 分段函数
A piecewise function is defined by different rules on different intervals. A common example is the absolute value function. Piecewise functions are used to model real-world situations with changing rates or rules.
分段函数在不同区间上由不同的表达式定义。绝对值函数是常见例子。分段函数常用于建模具有变化规则或变化速率的现实情境。
- To evaluate, first determine which interval contains the input.
- 求值时先判断输入值属于哪一区间。
- Check continuity and differentiability at the interval boundaries.
- 在区间交界处需检查连续性与可导性。
- A step function (e.g., floor function) is a special type of piecewise function.
- 阶梯函数（如取整函数）是分段函数的特例。
11. Inverse Functions | 反函数
If f is one-to-one, its inverse f⁻¹ satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x graph of f⁻¹ is the reflection of the graph of f in the line y = x. Domain and range swap: Dom(f⁻¹) = Range(f), and vice versa.
如果 f 是单射，则其反函数 f⁻¹ 满足 f(f⁻¹(x)) = x 与 f⁻¹(f(x)) = x。f⁻¹ 的图像是 f 的图像关于直线 y = x 的反射。定义域与值域互换：Dom(f⁻¹) = Range(f)，反之亦然。
- Not every function has an inverse; it must be one-to-one.
- 并非所有函数都有反函数，要求必须是单射。
- To find f⁻¹, swap x and y in y = f(x), then solve for y.
- 求 f⁻¹ 的方法：在 y = f(x) 中交换 x 与 y，再解出 y。
- Common inverses: exponential ↔ logarithmic; quadratic (with restricted domain) ↔ square root.
- 常见反函数：指数函数 ↔ 对数函数；二次函数（限制定义域后）↔ 平方根函数。
12. Hyperbolic Functions | 双曲函数
Hyperbolic functions are defined using exponential functions: sinh x = (eˣ - e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x / cosh x. They appear in calculus and applications such as catenary curves. Their inverses involve logarithms.
双曲函数由指数函数定义：sinh x = (eˣ - e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2，tanh x = sinh x / cosh x。它们出现在微积分以及悬链线等应用中。其反函数与对数有关。
- cosh x is even; sinh x and tanh x are odd.
- cosh x 为偶函数；sinh x 与 tanh x 为奇函数。
- Identity: cosh²x - sinh²x = 1.
- 恒等式：cosh²x - sinh²x = 1。
- Derivatives: d/dx(sinh x) = cosh x; d/dx(cosh x) = sinh x.
- 导数：d/dx(sinh x) = cosh x；d/dx(cosh x) = sinh x。
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