Understanding Functions and Common Exam Points | 函数概念的理解与常见考点

📚 Understanding Functions and Common Exam Points | 函数概念的理解与常见考点

Functions are one of the most fundamental concepts in IB Mathematics. Understanding the underlying idea of a function, its notation, and its behaviour is essential for solving exam problems across all topics.

函数是IB数学中最基本的概念之一。理解函数的基本思想、记号及其行为，对于解答各主题下的考试题目至关重要。

1. What is a Function? | 什么是函数？

In simple terms, a function is a rule that assigns each input exactly one output.

简单来说，函数是一种规则，它把每一个输入值唯一地对应到一个输出值。

For example, f(x) = x² + 1 takes an input x and gives the output x² + 1.

例如，f(x) = x² + 1 接收输入 x，并给出输出 x² + 1。

Mathematically, a function from set A to set B is a relation where every element of A is paired with exactly one element of B.

从数学上看，从集合A到集合B的函数是一种关系，其中A中的每个元素都恰好与B中的一个元素对应。

The vertical line test is a quick way to decide whether a curve is a function: if any vertical line crosses the graph more than once, the graph does not represent a function.

垂直线检验法是判断曲线是否为函数的快捷方法：如果任何一条垂直线与图像相交超过一次，该图像就不表示函数。

2. Domain, Codomain and Range | 定义域、陪域与值域

Three important words describe the sets involved in a function.

有三个重要术语描述函数所涉及的集合。

Domain: the set of all allowed input values.

定义域：所有允许的输入值的集合。
Codomain: the set that contains the possible outputs.

陪域：包含所有可能输出值的集合。
Range: the set of actual outputs produced by the function.

值域：函数实际产生的输出值的集合。

For example, for f(x) = √(x − 2), the domain is x ≥ 2, written as x ∈ [2, ∞). The range is also y ≥ 0.

例如，对于 f(x) = √(x − 2)，定义域为 x ≥ 2，即 x ∈ [2, ∞)。值域为 y ≥ 0。

In IB exams, you are often asked to find the domain and range of a function from its graph or its algebraic expression.

在IB考试中，经常要求你根据函数图像或代数表达式求定义域和值域。

3. Function Notation and Evaluation | 函数记号与求值

The symbol f(x) is read as “f of x”. It tells us which rule to apply to x.

符号 f(x) 读作 “f of x”。它告诉我们对 x 应用什么规则。

To evaluate a function at a specific value, substitute that value into the expression.

要在特定值处求函数值，只需将该值代入表达式中。

For example, if f(x) = 2x − 1, then f(3) = 2(3) − 1 = 5.

例如，若 f(x) = 2x − 1，则 f(3) = 2(3) − 1 = 5。

You may also see function notation with different letters, such as g(t) or h(x), and the same rules apply.

你也可能看到用不同字母表示的函数记号，如 g(t) 或 h(x)，规则完全相同。

4. Composite Functions | 复合函数

A composite function applies one function after another.

复合函数是一个函数接着另一个函数应用。

The notation (f ∘ g)(x) means f(g(x)). Always work from the inside out.

记号 (f ∘ g)(x) 表示 f(g(x))。运算时总是由内向外进行。

(f ∘ g)(x) = f(g(x))

For example, let f(x) = x² and g(x) = x + 1. Then (f ∘ g)(x) = f(g(x)) = (x + 1)², while (g ∘ f)(x) = g(f(x)) = x² + 1.

例如，设 f(x) = x²，g(x) = x + 1。则 (f ∘ g)(x) = f(g(x)) = (x + 1)²，而 (g ∘ f)(x) = g(f(x)) = x² + 1。

Notice that order matters: f ∘ g is usually not equal to g ∘ f.

注意顺序很重要：f ∘ g 通常不等于 g ∘ f。

When forming a composite function, the range of the inner function must be compatible with the domain of the outer function.

构成复合函数时，内层函数的值域必须与外层函数的定义域相容。

5. Inverse Functions | 反函数

An inverse function reverses the effect of the original function.

反函数抵消原函数的作用。

If f(a) = b, then f⁻¹(b) = a. The inverse exists only when the original function is one-to-one (injective).

若 f(a) = b，则 f⁻¹(b) = a。只有当原函数是一一对应（单射）时，反函数才存在。

f⁻¹(f(x)) = x and f(f⁻¹(x)) = x

To find the inverse algebraically, swap x and y in the equation y = f(x), then solve for y.

用代数方法求反函数时，在方程 y = f(x) 中交换 x 和 y，然后解出 y。

Graphically, the graph of f⁻¹ is the reflection of the graph of f in the line y = x.

在图像上，f⁻¹ 的图像是 f 的图像关于直线 y = x 的反射。

For example, if f(x) = 2x + 3, then y = 2x + 3; swapping gives x = 2y + 3, so y = (x − 3)/2, and therefore f⁻¹(x) = (x − 3)/2.

例如，若 f(x) = 2x + 3，则 y = 2x + 3；交换得 x = 2y + 3，所以 y = (x − 3)/2，因此 f⁻¹(x) = (x − 3)/2。

6. Graph Transformations | 图像变换

Transformations move or stretch the graph of a function without changing its basic shape.

变换会移动或拉伸函数图像，但不会改变其基本形状。

Translation: y = f(x − a) shifts the graph a units to the right; y = f(x) + b shifts it b units up.

平移：y = f(x − a) 将图像向右平移 a 个单位；y = f(x) + b 将其向上平移 b 个单位。
Reflection: y = −f(x) reflects across the x-axis; y = f(−x) reflects across the y-axis.

反射：y = −f(x) 关于 x 轴对称；y = f(−x) 关于 y 轴对称。
Stretch: y = af(x) stretches vertically by factor a; y = f(bx) compresses horizontally if b > 1.

伸缩：y = af(x) 在竖直方向拉伸 a 倍；y = f(bx) 在 b > 1 时水平压缩。

In IB questions, you must know how to describe these transformations in words, using terms such as “translate”, “reflect” and “stretch”.

在IB题目中，你必须会用 “平移” “反射” “拉伸” 等词语描述这些变换。

7. Finding Domain and Range: Exam Strategies | 求定义域与值域：考试策略

For rational functions like f(x) = 1/(x − 3), exclude values that make the denominator zero: the domain is x ≠ 3.

对于有理函数 f(x) = 1/(x − 3)，要排除使分母为零的值：定义域为 x ≠ 3。

For functions with even roots, the radicand must be non-negative: f(x) = √(x + 5) has domain x ≥ −5.

对于偶次根式函数，根号内的表达式必须非负：f(x) = √(x + 5) 的定义域为 x ≥ −5。

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Understanding Functions and Common Exam Points | 函数概念的理解与常见考点