📚 Domain and Range | 定义域与值域
In IB Mathematics, understanding the domain and range of a function is fundamental to analysing relationships between variables. The domain tells us all the permissible inputs, while the range describes every possible output. Mastering this topic not only helps with graph sketching but is also essential for calculus, transformations, and modelling real-world scenarios.
在 IB 数学中，理解函数的定义域和值域是分析变量之间关系的基础。定义域告诉我们所有可允许的输入，而值域则描述了每一个可能的输出。掌握这一主题不仅有助于绘制函数图像，对于微积分、图像变换以及现实情境建模也至关重要。
1. What is Domain and Range? | 什么是定义域与值域？
The domain of a function f is the set of all real numbers x for which f(x) is defined and yields a real output. The range is the set of all possible values that f(x) can take as x runs through the entire domain. We often denote the domain as D and the range as R.
函数 f 的定义域是使得 f(x) 有意义且输出为实数的全体实数 x 的集合。值域是当 x 取遍定义域时，f(x) 所能取到的所有可能值的集合。我们通常用 D 表示定义域，R 表示值域。
ts (range). For example, f(x) = 1/x cannot accept x = 0, so 0 is excluded from the domain.
可以把函数想象成一台机器：只能投放有效的输入（定义域），而机器只会产生特定的输出（值域）。例如，f(x) = 1/x 不能接受 x = 0，因此 0 被排除在定义域之外。
2. Notation: Set Builder and Interval | 表示法：集合描述法与区间表示
Domains and ranges can be expressed in set-builder notation, such as {x ∈ ℝ | arentheses for open endpoints (not included) and brackets for closed endpoints (included).
定义域和值域可以用集合描述法表示，如 {x ∈ ℝ | x ≥ 2}，或用区间表示法，如 [2, ∞)。区间表示法用小括号表示开端点（不包含），用方括号表示闭端点（包含）。
The table below summarises the most common types of intervals:
下表总结了最常见的区间类型：
| Inequality | Set-builder | Interval |
|---|---|---|
| x > a | {x | x > a} | (a, ∞) |
| x ≥ a | {x | x ≥ a} | [a, ∞) |
| x < b | {x | x < b} | (-∞, b) |
| x ≤ b | {x | x ≤ b} | (-∞, b] |
| a < x < b | {x | a < x < b} | (a, b) |
Unions are used when the domain splits into separate intervals, e.g., (-∞, 0) ∪ (0, ∞) for all real numbers except zero.
当定义域分裂成几个不连续的区间时，使用并集符号 ∪，例如除去零以外的全体实数可表示为 (-∞, 0) ∪ (0, ∞)。
3. Finding Domain from Equations | 从方程求定义域
When a function is given as an equation without a stated domain, we assume the domain is the largest subset of ℝ for which the expression is defined. Three main restrictions must be checked: denominators cannot be zero, even-index radicals must have non-negative radicands, and logarithmic arguments must be strictly positive.
当函数以方程形式给出且未指定定义域时，我们假定定义域是使得表达式有意义的最大实数子集。必须检查三大限制：分母不能为零，偶次根式的被开方数必须非负，对数函数的真数必须严格大于零。
For instance, to find the domain of f(x) = √x - 1 we require x - 1 ≥ 0 and x - 4 ≠ 0. Solving gives x ≥ 1 and x ≠ 4, so the domain is [1, 4) ∪ (4, ∞).
例如，求 f(x) = √x - 1 / (x - 4) 的定义域，需要 x - 1 ≥ 0 且 x - 4 ≠ 0。解不等式得 x ≥ 1 且 x ≠ 4，因此定义域为 [1, 4) ∪ (4, ∞)。
No restriction applies to odd roots, such as cube roots, or to terms like eˣ, where the exponent can be any real number.
对于奇次方根（如立方根）或 eˣ 这样的项，指数可以是任意实数，不存在限制。
4. Domain of Polynomial Functions | 多项式函数的定义域
ecause they involve only addition, subtraction, and multiplication. Hence, the domain of any polynomial is ℝ, or (-∞, ∞).
多项式函数形如 aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₀，其中 n 为非负整数。由于只涉及加、减和乘法运算，它们在全体实数上都有定义。因此，任何多项式的定义域都是 ℝ，即 (-∞, ∞)。
) = 4x³ - 5x + 2, g(x) = x² + 1, and even constant functions like h(x) = 7 all have domain ℝ. No matter what real number you substitute, you obtain a real result.
例如：f(x) = 4x³ - 5x + 2、g(x) = x² + 1，甚至常数函数 h(x) = 7 的定义域都是 ℝ。无论代入哪个实数，都会得到一个实数结果。
5. Domain of Rational Functions | 有理函数的定义域
ndefined. Therefore, we set the denominator equal to zero, solve for x, and remove those values from ℝ.
有理函数是两个多项式的比，f(x) = P(x)/Q(x)。它的定义域要除去使 Q(x) = 0 的 x 值，因为除以零无意义。因此，令分母为零，解出 x，并将这些值从 ℝ 中剔除。
minator factors as (x - 3)(x + 3). The restrictions are x ≠ 3 and x ≠ -3. The domain is {x ∈ ℝ | x ≠ ±3} or (-∞, -3) ∪ (-3, 3) ∪ (3, ∞).
对于 f(x) = (x + 2) / (x² - 9)，分母可分解为 (x - 3)(x + 3)。限制条件为 x ≠ 3 且 x ≠ -3。定义域为 {x ∈ ℝ | x ≠ ±3} 或 (-∞, -3) ∪ (-3, 3) ∪ (3, ∞)。
(x² - 1) = 1/(x + 1) after cancelling (x - 1), the domain still excludes x = 1 and x = -1.
即使化简时约去了某个公因式，原本的限制依然保留。对于 h(x) = (x - 1)/(x² - 1)，约去 (x - 1) 后等于 1/(x + 1)，但其定义域仍然排除 x = 1 和 x = -1。
6. Domain of Radical Functions | 根式函数的定义域
For a function involving an even root, such as f(x) = √ax + b olving this inequality gives the domain. If the radical appears in the denominator, the inequality becomes strictly greater than zero: ax + b > 0.
对于含偶次根式的函数，如 f(x) = √ax + b，被开方数必须满足 ax + b ≥ 0。解此不等式即得定义域。若根式出现在分母中，则不等式变为严格大于零：ax + b > 0。
Consider g(x) = √2x - 5. We require 2x - 5 ≥ 0 ⇒ x ≥ 5/2, so the domain is [2.5, ∞). For h(x) = 1/√4 - x, we need 4 - x > 0 ⇒ x < 4, giving domain (-∞, 4).
考虑 g(x) = √2x - 5，要求 2x - 5 ≥ 0 ⇒ x ≥ 5/2，因此定义域为 [2.5, ∞)。对于 h(x) = 1/√4 - x，需要 4 - x > 0 ⇒ x < 4，定义域为 (-∞, 4)。
Odd roots, such as cube roots, do not impose sign restrictions. Thus, f(x) = ∛(x - 2) has domain ℝ.
奇次方根（如立方根）没有符号限制。因此 f(x) = ∛(x - 2) 的定义域为 ℝ。
7. Domain of Logarithmic Functions | 对数函数的定义域
(x), and in particular the natural logarithm ln(x), is defined only for positive arguments. This means if the argument is an expression u(x), we must solve u(x) > 0 to find the domain.
对数函数 f(x) = logₐ(x)，尤其是自然对数 ln(x)，只对正的真数有定义。这意味着如果真数是一个表达式 u(x)，我们必须解 u(x) > 0 来求定义域。
Example: f(x) = ln(3x - 9). Set 3x - 9 > 0 ⇒ x), the argument must be positive (x > 0) and the denominator cannot be zero, so ln(x) ≠ 0 ⇒ x ≠ 1. The domain is (0, 1) ∪ (1, ∞).
例题：f(x) = ln(3x - 9)。令 3x - 9 > 0 得 x > 3。定义域：(3, ∞)。如果对数在分母中，例如 k(x) = 5 / ln(x)，真数必须为正 (x > 0)，且分母不可为零，故 ln(x) ≠ 0 ⇒ x ≠ 1。定义域为 (0, 1) ∪ (1, ∞)。
A common mistake is to forget that the argument itself may be a composite expression requiring further restrictions, always work from the inside out.
常见错误是忘记真数本身可能是复合表达式，需要额外的限制条件，务必从内到外依次处理。
8. Domain of Trigonometric Functions | 三角函数的定义域
cosine functions, sin x and cos x, are defined for all real numbers, so their domains are ℝ. However, the tangent function, tan x = sin x / cos x, is undefined whenever cos x = 0.
正弦函数 sin x 和余弦函数 cos x 对全体实数都有定义，因此它们的定义域为 ℝ。然而，正切函数 tan x = sin x / cos x 在 cos x = 0 时无定义。
cos x = 0 occurs at odd multiples of π/2, i.e., x = π/2 + kπ, k ∈ ℤ. Thus, the domain of tan x is {x ∈ ℝ | cant, and cotangent functions have similar domain restrictions based on their definitions.
cos x = 0 发生在 π/2 的奇数倍处，即 x = π/2 + kπ, k ∈ ℤ。因此，tan x 的定义域为 {x ∈ ℝ | x ≠ π/2 + kπ}。正割、余割和余切函数根据各自的定义也有类似的限制。
are functions. For instance, y = arcsin x has domain [-1, 1], and y = arctan x has domain ℝ.
对于反三角函数，为了使其成为函数，定义域被人为限制。例如，y = arcsin x 的定义域为 [-1, 1]，而 y = arctan x 的定义域为 ℝ。
9. Determining Range from Graphs | 从图像确定值域
One powerful way to find the range is to examine the graph of the function. The range is the projection of all points on the curve onto the y-axis. Look for the lowest and highest y-values reached, and consider whether the function approaches any horizontal asymptotes or has gaps.
求值域的一个有力方法是观察函数图像。值域是曲线上所有点向 y 轴投影所得的集合。寻找图像达到的最低和最高 y 值，并考虑函数是否趋近于水平渐近线或存在间断。
ily large as x → 0, while as x → ±∞, y → 0 but never reaches 0. Hence the range is (0, ∞).
例如，f(x) = 1/x² 的图像显示 y 恒为正值，当 x → 0 时可无限增大，而当 x → ±∞ 时 y → 0 但永远不等于 0。因此值域为 (0, ∞)。
When reading graphs, pay attention to solid dots (included) versus open circles (excluded), as these indicate whether an endpoint belongs to the range.
解读图像时，注意实心圆点（包含）与空心圆圈（不包含）的区别，它们表明端点是否属于值域。
10. Range of Linear and Quadratic Functions | 一次函数和二次函数的值域
A linear function f(x) = mx + c with m ≠ 0 covers all real numbers as x runs through ℝ, so its range is ℝ. If m = 0, the function is constant and the range is the single value {c}.
一次函数 f(x) = mx + c（m ≠ 0）在 x 取遍 ℝ 时会覆盖全体实数，因此值域为 ℝ。若 m = 0，函数为常值函数，值域为单点集 {c}。
x + c (a ≠ 0), the graph is a parabola. The vertex is the turning point, and its y-coordinate, often denoted k, determines the boundary of the range.
对于二次函数 f(x) = ax² + bx + c (a ≠ 0)，其图像为抛物线。顶点为转折点，其 y 坐标（常记作 k）决定了值域的边界。
k = (4ac − b²) / (4a)
< 0, the parabola opens downward and the vertex is a maximum, yielding the range (-∞, k].
若 a > 0，抛物线开口向上，顶点为最小值点，值域为 [k, ∞)。若 a < 0，抛物线开口向下，顶点为最大值点，值域为 (-∞, k]。
oordinate is k = (4(-2)(-3) - 8²) / (4(-2)) = (24 - 64)/(-8) = 5, so the range is (-∞, 5].
对于 f(x) = -2x² + 8x - 3，a = -2，b = 8，c = -3。顶点的 y 坐标为 k = (4(-2)(-3) - 8²) / (4(-2)) = (24 - 64)/(-8) = 5，因此值域为 (-∞, 5]。
11. Range of Exponential, Logarithmic and Other Functions | 指数、对数及其他函数的值域
Exponential functions of the form f(x) = a · bˣ b ≠ 1) have a horizontal asymptote y = d. For b > 1 and a > 0, the function grows without
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