Restrictions on the Value of x | 变量x取值的限制

📚 Restrictions on the Value of x | 变量x取值的限制

In A-Level mathematics, we often need to determine which values of a variable are allowed in a given expression or equation. These allowed values form the domain, and any value that makes the expression undefined or invalid is excluded. Understanding these restrictions is essential for solving equations, sketching graphs, and modelling real-life problems.

在A-Level数学中，我们经常需要确定变量在给定表达式或方程中被允许取哪些值。这些允许的取值构成定义域，而使表达式无意义或无效的值则被排除。理解这些取值限制对于解方程、绘制图像以及建立现实问题模型都至关重要。

1. What Does “Restriction” Mean? | “限制”的含义

A restriction on the value of x is a condition that x must satisfy so that a mathematical expression or function is well-defined. For example, the expression 1/(x − 2) is defined for every real x except x = 2; we say the restriction is x ≠ 2.

对x取值的限制是指x必须满足的条件，使得某个数学表达式或函数有定义。例如，表达式1/(x − 2)对所有实数x都有定义，除了x = 2；我们就说其限制为x ≠ 2。

Typical sources of restrictions include:

限制的常见来源包括：

Even roots (square roots, fourth roots, etc.) require the radicand to be non-negative.
偶次根（平方根、四次根等）要求被开方数为非负数。
Denominators cannot be zero.
分母不能为零。
Logarithms require the argument to be strictly positive.
对数要求真数严格为正。
Inverse trigonometric functions (arcsin, arccos) have restricted input intervals.
反三角函数（arcsin、arccos）的输入有区间限制。

2. Restrictions from Square Roots and Even Roots | 平方根与偶次根的限制

For a square root, the expression inside the root must be greater than or equal to zero. In general, for any even root, the radicand must be non-negative.

对于平方根，根号内的表达式必须大于或等于零。一般地，对于任何偶次根，被开方数必须是非负数。

√(f(x)) is defined only when f(x) ≥ 0

√(f(x)) 仅在 f(x) ≥ 0 时有定义

Example: Find the domain of f(x) = √(3 − x).

示例：求 f(x) = √(3 − x) 的定义域。

We require 3 − x ≥ 0, so x ≤ 3. The domain is {x ∈ ℝ : x ≤ 3}.

我们需要 3 − x ≥ 0，即 x ≤ 3。定义域为 {x ∈ ℝ : x ≤ 3}。

For the fourth root, such as (x² − 4)^(1/4), we need x² − 4 ≥ 0, which gives x ≤ −2 or x ≥ 2.

对于四次根，如 (x² − 4)^(1/4)，我们需要 x² − 4 ≥ 0，得到 x ≤ −2 或 x ≥ 2。

3. Restrictions from Denominators | 分母的限制

A rational expression is undefined when its denominator is equal to zero. Therefore, any value of x that makes the denominator zero must be excluded.

有理表达式在其分母等于零时无定义。因此，任何使分母为零的x值都必须被排除。

1 / g(x) is defined only when g(x) ≠ 0

1 / g(x) 仅在 g(x) ≠ 0 时有定义

Example: f(x) = 1/(x² − 5x + 6). Factorise the denominator: (x − 2)(x − 3). Thus x ≠ 2 and x ≠ 3.

示例：f(x) = 1/(x² − 5x + 6)。因式分解分母：(x − 2)(x − 3)。因此 x ≠ 2 且 x ≠ 3。

When a denominator contains a square root, both the denominator and the root impose restrictions. For example, 1/√(x − 1) requires x − 1 > 0 because √(x − 1) must be non-zero; hence x > 1.

当分母含有平方根时，分母和根号都会施加限制。例如，1/√(x − 1) 要求 x − 1 > 0，因为√(x − 1)必须非零；因此 x > 1。

4. Restrictions from Logarithms | 对数的限制

The logarithm function logₐ(x) is defined only when x > 0. This applies to any base a > 0, a ≠ 1, including the natural logarithm ln(x).

对数函数 logₐ(x) 仅在 x > 0 时有定义。这适用于任何底数 a > 0，a ≠ 1，包括自然对数 ln(x)。

ln(f(x)) is defined only when f(x) > 0

ln(f(x)) 仅在 f(x) > 0 时有定义

Example: Find the domain of g(x) = ln(2x + 6). We require 2x + 6 > 0, so x > −3.

示例：求 g(x) = ln(2x + 6) 的定义域。我们需要 2x + 6 > 0，所以 x > −3。

Be careful: the argument of a logarithm must be strictly positive, not just non-negative. A common mistake is writing ≥ 0 instead of > 0.

注意：对数的真数必须严格为正，而不是非负。常见错误是写成 ≥ 0 而不是 > 0。

5. Restrictions from Inverse Trigonometric Functions | 反三角函数的限制

The inverse trigonometric functions arcsin x and arccos x are defined only for x in the interval [−1, 1]. The function arctan x is defined for all real x.

反三角函数 arcsin x 和 arccos x 仅在 x 属于区间 [−1, 1] 时有定义。arctan x 对所有实数x都有定义。

arcsin x is defined only when −1 ≤ x ≤ 1

arcsin x 仅在 −1 ≤ x ≤ 1 时有定义

Example: f(x) = arcsin(2x). We require −1 ≤ 2x ≤ 1, hence −½ ≤ x ≤ ½.

示例：f(x) = arcsin(2x)。我们需要 −1 ≤ 2x ≤ 1，因此 −½ ≤ x ≤ ½。

Similarly, arccos(x − 1) requires −1 ≤ x − 1 ≤ 1, giving 0 ≤ x ≤ 2.

类似地，arccos(x − 1) 要求 −1 ≤ x − 1 ≤ 1，得到 0 ≤ x ≤ 2。

6. Restrictions from Other Standard Functions | 其他标准函数的限制

The tangent function tan x is undefined at x = π/2 + kπ, where k is any integer. This is because cos x = 0 at those points.

正切函数 tan x 在 x = π/2 + kπ 处无定义，其中k为任意整数。这是因为在这些点上cos x = 0。

Similarly, cot x = cos x / sin x is undefined when sin x = 0, i.e. x = kπ. Also, sec x = 1/cos x has the same restrictions as tan x, and cosec x = 1/sin x has the same restrictions as cot x.

类似地，cot x = cos x / sin x 在 sin x = 0 时无定义，即 x = kπ。另外，sec x = 1/cos x 与 tan x 有相同的限制，cosec x = 1/sin x 与 cot x 有相同的限制。

Exponential functions such as eˣ are defined for all real x; there is no restriction from the exponent itself, but if the exponent is a fraction or contains a logarithm, further conditions may appear.

指数函数如 eˣ 对所有实数x有定义；指数本身没有限制，但如果指数是分数或包含对数，则可能出现进一步的条件。

7. Combining Multiple Restrictions | 组合多个限制

When an expression contains several restricted parts, the overall domain is the intersection of all individual domains.

当一个表达式包含多个受限部分时，整体定义域是所有单个定义域的交集。

Example: Find the domain of h(x) = √(ln x).

示例：求 h(x) = √(ln x) 的定义域。

First, ln x is defined only when x > 0. Second, the square root requires ln x ≥ 0, which means x ≥ 1 because ln x ≥ 0 for x ≥ 1. Taking the intersection, the domain is x ≥ 1.

首先，ln x 仅在 x > 0 时有定义。其次，平方根要求 ln x ≥ 0，这意味着 x ≥ 1，因为当 x ≥ 1 时 ln x ≥ 0。取交集，定义域为 x ≥ 1。

Another example: f(x) = 1 / √(x² − 1). The denominator must not be zero, so x² − 1 > 0, hence x < −1 or x > 1. There is no additional restriction because the square root itself is already handled by the strict inequality.

另一个例子：f(x) = 1 / √(x² − 1)。分母不能为零，所以 x² − 1 > 0，因此 x < −1 或 x > 1。由于严格不等式已经处理了根号，所以没有额外限制。

8. Solving Inequalities to Find Restrictions | 用不等式求限制

Many restrictions are expressed as inequalities. To find the domain, solve these inequalities carefully, especially when quadratic or rational expressions are involved.

许多限制以不等式形式表达。要求定义域，需要仔细解这些不等式，特别是当涉及二次或有理表达式时。

Example: Find the domain of f(x) = √(x² − x − 2).

示例：求 f(x) = √(x² − x − 2) 的定义域。

We need x² − x − 2 ≥ 0. Factorise: (x − 2)(x + 1) ≥ 0. Using a sign table or graph, the solution is x ≤ −1 or x ≥ 2.

我们需要 x² − x − 2 ≥ 0。因式分解：(x − 2)(x + 1) ≥ 0。通过符号表或图像，解为 x ≤ −1 或 x ≥ 2。

When multiplying or dividing an inequality by a negative number, reverse the inequality sign. This is a frequent source of error when solving for restrictions.

当不等式两边乘以或除以一个负数时，必须反转不等号。这是在求解限制时常见的错误来源。

9. Real-World Context and Domain Modelling | 现实背景与建模中的定义域

In applied mathematics, restrictions often come from the physical meaning of the variable. For example, time t is usually non-negative, distances are non-negative, and population counts are integers.

在应用数学中，限制通常来自变量的物理意义。例如，时间t通常是非负的，距离是非负的，人口数量是整数。

Suppose a projectile’s height is given by h(t) = 20t − 5t². Although the quadratic expression is defined for all t, a sensible domain for modelling the flight is 0 ≤ t ≤ 4, since h(t) ≥ 0 during flight.

假设一个抛射体的高度由 h(t) = 20t − 5t² 给出。虽然二次表达式对所有t都有定义，但模拟飞行的合理定义域是 0 ≤ t ≤ 4，因为在飞行期间 h(t) ≥ 0。

When building a model, always state the domain based on both mathematical validity and real-world constraints. The mathematical restrictions alone may not capture the full picture.

在建立模型时，始终要基于数学有效性和现实约束来写出定义域。仅靠数学限制可能无法反映全貌。

10. Exam Tips and Common Mistakes | 考试技巧与常见错误

In A-Level exams, domain questions are often worth a few marks. You must show the inequality or exclusion clearly and write the final domain in set notation or interval notation.

在A-Level考试中，定义域问题通常占几分。你必须清楚写出不等式或排除条件，并用集合记号或区间记号写出最终定义域。

Common mistakes to avoid:

需要避免的常见错误：

Forgetting that √(x) requires x ≥ 0, not x > 0 (except in denominators).
忘记√(x)要求 x ≥ 0，而不是 x > 0（在分母中除外）。
Using ≥ 0 for a logarithm; logarithms demand > 0.
对对数使用 ≥ 0；对数要求 > 0。
Dropping the condition x ≠ 0 when simplifying expressions like 1/(x(x−1)).
在化简如 1/(x(x−1)) 的表达式时忽略 x ≠ 0 的条件。
Solving inequalities incorrectly when multiplying by a negative number.
乘以负数时错误地解不等式。

Here is a quick reference table for common functions and their restrictions:

下表是常见函数及其限制的速查表：

Function / 函数	Restriction / 限制	Example / 示例
√(x)	x ≥ 0	√(x − 2) → x ≥ 2
1/x	x ≠ 0	1/(x+3) → x ≠ −3
ln(x)	x > 0	ln(1 − x) → x < 1
arcsin(x)	−1 ≤ x ≤ 1	arcsin(3x) → −⅓ ≤ x ≤ ⅓
tan(x)	x ≠ π/2 + kπ	tan(2x) → x ≠ π/4 + kπ/2

Always write your final answer in the form requested, for example “x > 3” or “{x ∈ ℝ : x ≤ 5}”. In AQA markschemes, both set notation and interval notation are accepted.

始终按要求的形式写出最终答案，例如 “x > 3” 或 “{x ∈ ℝ : x ≤ 5}”。在AQA评分标准中，集合记法和区间记法都被接受。

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Restrictions on the Value of x | 变量x取值的限制