Restrictions on the Values of a Function | 函数值的限制

📚 Restrictions on the Values of a Function | 函数值的限制

In AQA A-Level Mathematics, many functions cannot accept every real number as an input, and their outputs are often limited too. Understanding these restrictions is essential for sketching graphs, solving equations, finding inverses and working with composite functions.

在 AQA A-Level 数学中，很多函数并非对所有实数都有定义，其输出也常常受到限制。理解这些限制对绘制图像、解方程、求反函数以及处理复合函数都至关重要。

1. Why Restrictions Arise | 为什么会产生限制

A function is only well-defined when every allowed input produces exactly one real output. Restrictions appear when a formula would otherwise lead to division by zero, the square root of a negative number, the logarithm of a non-positive number, or other undefined operations.

只有当一个函数对每个允许的输入都能产生唯一实数输出时，它才是良定义的。当公式可能导致除以零、负数开平方、取非正数的对数或其他无定义运算时，限制就会出现。

In AQA exam questions, you are often asked to state the largest possible domain or the full range of a function, so you must learn to spot these restrictions quickly.

在 AQA 考试题中，经常要求写出函数的最大可能定义域或完整值域，因此你必须学会快速识别这些限制。

2. Domain Restrictions from Denominators | 分母带来的定义域限制

Division by zero is undefined, so any value of x that makes a denominator equal to zero must be excluded from the domain of a rational function.

除以零是无定义的，因此任何使分母为零的 x 值都必须从有理函数的定义域中排除。

For example, if f(x) = 1/(x – 3), the denominator is zero when x = 3, so the domain is all real numbers except x = 3. This can be written as {x ∈ ℝ : x ≠ 3} or (-∞, 3) ∪ (3, ∞).

例如，若 f(x) = 1/(x – 3)，当 x = 3 时分母为零，所以定义域是所有实数但 x ≠ 3。这可以记作 {x ∈ ℝ : x ≠ 3} 或 (-∞, 3) ∪ (3, ∞)。

For f(x) = (x + 2)/(x² – 4), factor the denominator: x² – 4 = (x – 2)(x + 2). Both x = 2 and x = -2 make the denominator zero, so exclude both values.

对于 f(x) = (x + 2)/(x² – 4)，将分母因式分解：x² – 4 = (x – 2)(x + 2)。x = 2 和 x = -2 都会使分母为零，因此都要排除。

3. Domain Restrictions from Square Roots | 平方根带来的定义域限制

For real functions, the expression inside a square root must be greater than or equal to zero. This is because the square root of a negative number is not a real number.

对于实函数，平方根内的表达式必须大于或等于零。这是因为负数的平方根不是实数。

If f(x) = √(x – 4), then we need x – 4 ≥ 0, so x ≥ 4. The domain is [4, ∞) or {x ∈ ℝ : x ≥ 4}.

若 f(x) = √(x – 4)，则需要 x – 4 ≥ 0，因此 x ≥ 4。定义域为 [4, ∞) 或 {x ∈ ℝ : x ≥ 4}。

For f(x) = √(9 – x²), solve 9 – x² ≥ 0. This gives -3 ≤ x ≤ 3, often written as [-3, 3].

对于 f(x) = √(9 – x²)，解不等式 9 – x² ≥ 0，得到 -3 ≤ x ≤ 3，常写作 [-3, 3]。

4. Domain Restrictions from Logarithms | 对数带来的定义域限制

The logarithm logₐ(x) is only defined for x > 0. The base a must also be positive and not equal to 1, but in A-Level questions the base is usually fixed, such as ln x or log₁₀ x.

对数 logₐ(x) 只有当 x > 0 时才有定义。底数 a 必须大于 0 且不等于 1，但在 A-Level 题目中底数通常固定，如 ln x 或 log₁₀ x。

For f(x) = ln(2x – 5), the argument must satisfy 2x – 5 > 0, so x > 2.5. The domain is (2.5, ∞).

对于 f(x) = ln(2x – 5)，真数必须满足 2x – 5 > 0，所以 x > 2.5。定义域为 (2.5, ∞)。

Remember that a logarithm can have any real output, so log functions do not restrict the range by themselves, only the domain.

请记住，对数的输出可以是任意实数，因此对数函数本身不限制值域，只限制定义域。

5. Range Restrictions and How to Find Them | 值域限制及其求法

The range of a function is the set of all possible output values. Restrictions in the range arise from the shape of the graph, particularly from squared terms, square roots, absolute values, and asymptotes.

函数的值域是所有可能输出值的集合。值域的限制来自图像的形状，尤其是平方项、平方根、绝对值以及渐近线。

For f(x) = x² + 3, since x² ≥ 0, we have f(x) ≥ 3. The range is [3, ∞).

对于 f(x) = x² + 3，因为 x² ≥ 0，所以 f(x) ≥ 3。值域为 [3, ∞)。

For f(x) = 1/x, the output can be any real number except 0, so the range is {y ∈ ℝ : y ≠ 0} or (-∞, 0) ∪ (0, ∞).

对于 f(x) = 1/x，输出可以是除 0 以外的任意实数，因此值域为 {y ∈ ℝ : y ≠ 0} 或 (-∞, 0) ∪ (0, ∞)。

Sketching the graph is usually the fastest way to identify the range, especially after you have found the domain.

绘制图像通常是确定值域的最快方法，尤其是在你已经求出定义域之后。

6. Using Set and Interval Notation | 使用集合与区间表示法

AQA expects you to express domains and ranges clearly. Interval notation uses brackets and parentheses: [a, b] means a ≤ x ≤ b, while (a, b) means a < x < b. A union symbol ∪ joins separate intervals.

AQA 要求你清晰表达定义域和值域。区间表示法使用方括号和圆括号：[a, b] 表示 a ≤ x ≤ b，而 (a, b) 表示 a < x < b。并集符号 ∪ 连接不连续的区间。

For example, the domain of f(x) = √(x + 2)/(x – 1) must satisfy x + 2 ≥ 0 and x – 1 ≠ 0. So x ≥ -2 and x ≠ 1. In interval notation: [-2, 1) ∪ (1, ∞).

例如，函数 f(x) = √(x + 2)/(x – 1) 的定义域必须满足 x + 2 ≥ 0 且 x – 1 ≠ 0。因此 x ≥ -2 且 x ≠ 1。用区间表示：[-2, 1) ∪ (1, ∞)。

Set-builder notation is also accepted, such as {x ∈ ℝ : x ≥ -2, x ≠ 1}. Make sure you do not accidentally include the excluded value.

集合构造式表示法也是可接受的，例如 {x ∈ ℝ : x ≥ -2, x ≠ 1}。确保不要不小心包含被排除的值。

7. Restrictions in Composite Functions | 复合函数中的限制

When you form a composite function such as fg(x) = f(g(x)), the input x must first belong to the domain of g, and the output g(x) must belong to the domain of f.

当你构造复合函数如 fg(x) = f(g(x)) 时，输入 x 首先必须属于 g 的定义域，并且输出 g(x) 必须属于 f 的定义域。

For example, let f(x) = √x and g(x) = x – 5. Then fg(x) = √(x – 5). The domain of g is all real numbers, but the domain of f requires x – 5 ≥ 0, so the composite domain is x ≥ 5.

例如，设 f(x) = √x，g(x) = x – 5。则 fg(x) = √(x – 5)。g 的定义域是所有实数，但 f 的定义域要求 x – 5 ≥ 0，因此复合函数的定义域为 x ≥ 5。

Always find the composite formula first, then work out the restrictions on the final expression, but also check the inner function’s domain.

总是先求出复合公式，然后确定最终表达式上的限制，同时也要检查内层函数的定义域。

8. Restrictions in Inverse Functions | 反函数中的限制

The domain of a function becomes the range of its inverse, and the range of the function becomes the domain of the inverse. This means restrictions swap roles.

函数的定义域成为其反函数的值域，函数的值域成为反函数的定义域。这意味着限制互换了角色。

For example, if f(x) = x² for x ≥ 0, then f has domain [0, ∞) and range [0, ∞). Its inverse f⁻¹(x) = √x also has domain [0, ∞) and range [0, ∞).

例如，若 f(x) = x² 且 x ≥ 0，则 f 的定义域为 [0, ∞)，值域为 [0, ∞)。其反函数 f⁻¹(x) = √x 的定义域也为 [0, ∞)，值域也为 [0, ∞)。

When a function is not one-to-one on its whole natural domain, you must restrict the domain to make an inverse possible, such as restricting y = x² to x ≥ 0.

当函数在其整个自然定义域上不是一一对应时，必须限制定义域才能存在反函数，例如将 y = x² 限制为 x ≥ 0。

9. Common Exam Mistakes | 常见考试错误

One common error is forgetting to exclude denominator zeros after simplifying a rational function. Simplify first, but still record any excluded values from the original expression.

一个常见错误是在化简有理函数后忘记排除分母为零的值。要先化简，但仍要记录原表达式中被排除的值。

Another error is writing the range of a quadratic as all real numbers. A

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Restrictions on the Values of a Function | 函数值的限制