Finding Domain and Range of Functions | 函数的定义域与值域求法

📚 Finding Domain and Range of Functions | 函数的定义域与值域求法

In IB Mathematics, finding the domain and range of a function is one of the most basic yet frequently tested skills. It appears in questions about graphs, composite functions, inverse functions and real-world modelling.

在 IB 数学中，求函数的定义域与值域是最基础却又极其常见的考点，它常与图像、复合函数、反函数以及实际建模问题结合考查。

1. Domain and Range: Definitions | 定义域与值域的定义

For a function f, the domain is the set of all allowed input values x, and the range is the set of all possible output values f(x). In IB notation we often write f: x ↦ f(x), with the domain shown explicitly, or implied by the algebraic expression.

对于函数 f，定义域是允许输入的全体 x 所构成的集合，而值域是这些输入所对应的全部输出 f(x) 的集合。在 IB 记号中，我们常写作 f: x ↦ f(x)，定义域由表达式明确给出或隐含确定。

Example: for f(x) = x², the domain is all real numbers x ∈ ℝ, and the range is y ≥ 0.

例如：对 f(x) = x²，定义域是全体实数 x ∈ ℝ，值域是 y ≥ 0。

2. Core Restrictions on the Domain | 定义域的基本限制

For any function, the domain is the set of x-values for which the algebraic expression is defined. The most important restrictions are below.

对任何函数而言，定义域是使代数表达式有意义的 x 的集合。最重要的限制如下。

Denominator cannot be zero: if f(x) has a fraction, solve denominator ≠ 0.

分母不能为零：若 f(x) 含有分式，则需要令分母 ≠ 0。
Even-index radicals must have a non-negative radicand: if f(x) = √g(x), require g(x) ≥ 0.

偶次根号内的式子必须非负：若 f(x) = √g(x)，则要求 g(x) ≥ 0。
Logarithms require a positive argument: if f(x) = logₐ g(x), require g(x) > 0.

对数要求真数为正：若 f(x) = logₐ g(x)，则要求 g(x) > 0。
Tangent functions are undefined at x = π/2 + kπ, where k ∈ ℤ.

正切函数在 x = π/2 + kπ 处无定义，其中 k ∈ ℤ。

3. Polynomial Functions | 多项式函数

Any polynomial function p(x) has domain ℝ. The range depends on its degree and leading coefficient.

任何多项式函数 p(x) 的定义域都是 ℝ。值域由其次数和首项系数决定。

A linear function f(x) = ax + b with a ≠ 0 has range ℝ.

一次函数 f(x) = ax + b（a ≠ 0）的值域是 ℝ。

A quadratic function f(x) = ax² + bx + c has a restricted range. Complete the square to find the maximum or minimum value.

二次函数 f(x) = ax² + bx + c 的值域是受限的，可通过配方法求出最大值或最小值。

Example: f(x) = x² – 4x + 3 = (x – 2)² – 1, so its domain is ℝ and its range is y ≥ -1.

例：f(x) = x² – 4x + 3 = (x – 2)² – 1，因此定义域为 ℝ，值域为 y ≥ -1。

4. Rational Functions | 有理函数

For a rational function, the domain excludes x-values that make the denominator zero. The range often excludes the horizontal asymptote value.

对有理函数而言，定义域要排除使分母为零的 x 值；值域常常要排除水平渐近线所对应的 y 值。

Example: consider f(x) = 2/(x + 1). The denominator x + 1 = 0 gives x = -1, so the domain is x ∈ ℝ, x ≠ -1.

例：考虑 f(x) = 2/(x + 1)。由分母 x + 1 = 0 得 x = -1，所以定义域为 x ∈ ℝ，x ≠ -1。

To find the range, solve the equation for x in terms of y:

求值域时，将原式改写为用 y 表示 x：

y = 2/(x + 1) → x = 2/y – 1

Since division by zero is not allowed, y ≠ 0, so the range is y ∈ ℝ, y ≠ 0.

由于不能除以零，所以 y ≠ 0，因此值域为 y ∈ ℝ，y ≠ 0。

5. Radical Functions | 根式函数

For even-index radicals such as square roots, we must solve radicand ≥ 0. For odd-index radicals such as cube roots, the domain is usually all real numbers.

对于平方根这类偶次根式，必须解被开方数 ≥ 0；对于立方根这类奇次根式，定义域通常是全体实数。

Example 1: f(x) = √(2x – 6). The radicand gives 2x – 6 ≥ 0, so x ≥ 3, and the range is y ≥ 0.

例 1：f(x) = √(2x – 6)。由被开方数可得 2x – 6 ≥ 0，故 x ≥ 3，值域为 y ≥ 0。

Example 2: f(x) = √(2x – 6) + 1. The domain is still x ≥ 3, and the range becomes y ≥ 1, because the smallest output is √0 + 1 = 1.

例 2：f(x) = √(2x – 6) + 1。定义域仍为 x ≥ 3，最小值 √0 + 1 = 1，所以值域变为 y ≥ 1。

Example 3: f(x) = ∛(x + 2) has domain ℝ and range ℝ.

例 3：f(x) = ∛(x + 2) 的定义域为 ℝ，值域也为 ℝ。

6. Exponential Functions | 指数函数

An exponential function of the form f(x) = aˣ with a > 0 and a ≠ 1 has domain ℝ and range y > 0. Vertical shifts change the range.

形如 f(x) = aˣ（a > 0 且 a ≠ 1）的指数函数，其定义域为 ℝ，值域为 y > 0。垂直平移会改变值域。

Example: f(x) = 2ˣ has domain ℝ and range y > 0.

例：f(x) = 2ˣ 的定义域为 ℝ，值域为 y > 0。

Example: f(x) = 3²ˣ⁻¹ + 2 has domain ℝ. Since 3²ˣ⁻¹ > 0, the range is y > 2.

例：f(x) = 3²ˣ⁻¹ + 2 的定义域为 ℝ。因为 3²ˣ⁻¹ > 0，所以值域为 y > 2。

Remember: the horizontal asymptote gives the excluded boundary of the range.

注意：水平渐近线对应的 y 值就是值域的边界，并且该值通常被排除。

7. Logarithmic Functions | 对数函数

A logarithmic function f(x) = logₐ g(x) is defined only when g(x) > 0. Its range is all real numbers.

对数函数 f(x) = logₐ g(x) 只有在 g(x) > 0 时才有定义，其值域为全体实数。

Example: f(x) = log₃(x – 2). The argument gives x – 2 > 0, so the domain is x > 2, and the range is ℝ.

例：f(x) = log₃(x – 2)。由真数可得 x – 2 > 0，所以定义域为 x > 2，值域为 ℝ。

In general, the vertical asymptote is found by setting the argument equal to zero.

一般地，令真数等于零即可找到竖直渐近线。

8. Trigonometric Functions | 三角函数

For y = sin x and y = cos x, the domain is ℝ and the range is -1 ≤ y ≤ 1.

对于 y = sin x 与 y = cos x，定义域为 ℝ，值域为 -1 ≤ y ≤ 1。

For y = tan x, the domain excludes x = π/2 + kπ, and the range is ℝ.

对于 y = tan x，定义域排除 x = π/2 + kπ，值域为 ℝ。

Example: f(x) = 3 cos(x – π) + 2 has domain ℝ. Since -3 ≤ 3 cos(x – π) ≤ 3, the range is -1 ≤ y ≤ 5.

例：f(x) = 3 cos(x – π) + 2 的定义域为 ℝ。因为 -3 ≤ 3 cos(x – π) ≤ 3，所以值域为 -1 ≤ y ≤ 5。

9. Piecewise and Composite Functions | 分段函数与复合函数

For a piecewise function, the domain is the union of all interval restrictions given in the definition. The range is the union of the output sets on each piece.

对于分段函数，定义域是各段区间限制的并集；值域是各段输出范围的并集。

For a composite function f ∘ g, we must first ensure that x is in the domain of g, and then ensure that g(x) is in the domain of f.

对于复合函数 f ∘ g，首先要保证 x 在 g 的定义域内，其次要保证 g(x) 在 f 的定义域内。

Example: f(x) = √x and g(x) = 1/(x – 2). Then (f ∘ g)(x) = √(1/(x – 2)). We need x ≠ 2 and also 1/(x – 2) ≥ 0, so the domain is x > 2.

例：f(x) = √x，g(x) = 1/(x – 2)。则 (f ∘ g)(x) = √(1/(x – 2))。需要 x ≠ 2 且 1/(x – 2) ≥ 0，因此定义域为 x > 2。

10. General Methods for Finding the Range | 求值域的常用方法

There is no single formula for the range, but the following techniques are the most useful in IB questions.

求值域没有万能公式，但以下方法在 IB 题目中最常用。

Inspection and graph sketching: draw the parent curve and apply transformations.

观察法与图像法：画出母函数图像并应用平移、伸缩等变换。
Completing the square: for quadratics, rewrite as a(x – h)² + k to find the vertex and the range boundary.

配方法：对二次函数改写为 a(x – h)² + k，从而确定顶点和值域边界。
Discriminant method: for rational functions, treat y as a constant and require the resulting quadratic in x to have real solutions.

判别式法：对有理函数，把 y 看作常数，令关于 x 的二次方程有实数解。
Monotonicity: if a function is increasing or decreasing on its domain, the range can be determined from the endpoints or asymptotes.

单调性法：若函数在定义域内递增或递减，则可根据端点或渐近线确定值域。

11. Transformations and Inverse Functions | 图象变换与反函数法

Vertical transformations shift the range, while horizontal transformations do not change the range.

垂直变换会平移值域，水平变换不会改变值域。

Example: if f(x) has range y ≥ 0, then f(x – 2) still has range y ≥ 0, but f(x) + 3 has range y ≥ 3.

例：若 f(x) 的值域为 y ≥ 0，则 f(x – 2) 的值域仍为 y ≥ 0，而 f(x) + 3 的值域变为 y ≥ 3。

If a function is one-to-one, the range of f equals the domain of f⁻¹. This is especially useful when the inverse is easier to analyse.

若函数是一一对应的，则 f 的值域等于 f⁻¹ 的定义域。当反函数更容易分析时，这一方法尤其好用。

12. Common Mistakes and Exam Tips | 常见错误与考试提示

A common mistake is to include values that make the denominator zero or make a logarithm argument zero. Always check asymptotes and endpoints.

常见错误是漏掉使分母为零或使对数真数为零的值。务必检查渐近线和端点。

Another common mistake is writing open and closed intervals incorrectly. Use round brackets for excluded values and square brackets for included values.

另一个常见错误是混淆开区间与闭区间。排除的值使用圆括号，包含的值使用方括号。

In IB exams, present the domain and range in set notation or interval notation, and justify your answer with a brief algebraic or graphical reason.

在 IB 考试中，请用集合记号或区间记号写出定义域与值域，并用简短的代数或图像理由说明结论。

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Finding Domain and Range of Functions | 函数的定义域与值域求法