📚 Functions: Concepts and Basic Properties | 函数概念与基本性质
A function is one of the most fundamental ideas in mathematics. It describes a special relationship between two sets: every input has exactly one output. In the IB Mathematics curriculum, a clear understanding of functions is essential for calculus, algebra, and modelling.
函数是数学中最基本的概念之一。它描述了两个集合之间的一种特殊关系：每一个输入都恰好对应一个输出。在 IB 数学课程中，清晰理解函数概念对微积分、代数和建模都至关重要。
1. Definition of a Function | 函数的定义
A function \(f\) from a set \(A\) to a set \(B\) assigns exactly one element of \(B\) to each element of \(A\). We write \(f: A \to B\), and the value of \(f\) at \(x\) is denoted \(f(x)\).
函数 \(f\) 从集合 \(A\) 到集合 \(B\)，对 \(A\) 中每个元素恰好指派 \(B\) 中一个元素。我们记为 \(f: A \to B\)，函数 \(f\) 在 \(x\) 处的值记为 \(f(x)\)。
The key condition is uniqueness: an input cannot produce two different outputs. For example, \(y = x^2\) is a function, but \(y^2 = x\) is not, because a single positive \(x\) gives two possible \(y\)-values.
关键条件是唯一性：一个输入不能产生两个不同的输出。例如，\(y = x^2\) 是函数，但 \(y^2 = x\) 不是，因为同一个正 \(x\) 会给出两个可能的 \(y\) 值。
| Function | 函数 | Not a function | 不是函数 |
| \(y = 2x + 1\) | \(y = \pm \sqrt{x}\) |
| \(y = \sin x\) | \(x^2 + y^2 = 1\) (full circle) |
2. Domain and Range | 定义域与值域
The domain is the set of all possible input values, while the range is the set of all possible output values. For a function \(f: A \to B\), the domain is \(A\), and the range is a subset of \(B\).
定义域是所有可能输入值的集合，而值域是所有可能输出值的集合。对于函数 \(f: A \to B\)，定义域是 \(A\)，值域是 \(B\) 的子集。
When the domain is not explicitly stated, we take the natural domain: all real numbers for which the expression is defined. For example, \(f(x) = \sqrt{x-2}\) has natural domain \(x \ge 2\), and its range is \(y \ge 0\).
当定义域没有明确给出时，我们取自然定义域：使表达式有定义的所有实数。例如，\(f(x) = \sqrt{x-2}\) 的自然定义域是 \(x \ge 2\)，其值域是 \(y \ge 0\)。
Common restrictions arise from square roots, logarithms, and denominators. For a rational function, the denominator cannot be zero; for a logarithmic function, the argument must be positive.
常见的限制来自平方根、对数和分母。对于分式函数，分母不能为零；对于对数函数，真数必须为正。
3. Function Notation and Evaluation | 函数记号与求值
Function notation \(f(x)\) is compact and powerful. To evaluate \(f(a)\), replace every occurrence of \(x\) in the expression with \(a\). For instance, if \(f(x) = x^2 + 3x\), then \(f(2) = 2^2 + 3(2) = 10\).
函数记号 \(f(x)\) 简洁而强大。要求 \(f(a)\)，只需将表达式中的每个 \(x\) 替换为 \(a\)。例如，若 \(f(x) = x^2 + 3x\)，则 \(f(2) = 2^2 + 3(2) = 10\)。
Nested evaluation is common: \(f(g(x))\) means first apply \(g\), then apply \(f\). This is exactly the idea of composition, which we discuss later.
嵌套求值很常见：\(f(g(x))\) 表示先应用 \(g\)，再应用 \(f\)。这正是复合函数的思想，我们稍后会讨论。
When working with piecewise functions, choose the correct expression based on the interval containing the input. Always verify interval endpoints carefully.
处理分段函数时，根据输入所在的区间选择正确的表达式。务必仔细检查区间端点。
4. Graphs of Functions | 函数的图像
The graph of a function \(f\) is the set of all points \((x, f(x))\) in the coordinate plane. It provides a visual representation of the relationship, making properties like intercepts and turning points easy to identify.
函数 \(f\) 的图像是坐标平面中所有点 \((x, f(x))\) 的集合。它提供了关系的直观表示，使截距和转折点等性质易于识别。
The vertical line test determines whether a curve represents a function: if any vertical line crosses the curve more than once, the curve is not a function.
垂线检验用于判断一条曲线是否表示函数：若任何一条垂直线与曲线相交多于一次，则该曲线不是函数。
Key features of a graph include \(x\)-intercepts (zeros), \(y\)-intercept, intervals of increase or decrease, and local maxima or minima. In IB questions, you are often asked to sketch or interpret these features.
图像的关键特征包括 \(x\) 截距（零点）、\(y\) 截距、递增或递减区间，以及局部最大值或最小值。在 IB 题目中，经常要求你画出或解释这些特征。
5. Composite Functions | 复合函数
Given two functions \(f\) and \(g\), the composite function \(f \circ g\) is defined by \((f \circ g)(x) = f(g(x))\). The domain of \(f \circ g\) consists of all \(x\) in the domain of \(g\) such that \(g(x)\) is in the domain of \(f\).
给定两个函数 \(f\) 和 \(g\)，复合函数 \(f \circ g\) 定义为 \((f \circ g)(x) = f(g(x))\)。\(f \circ g\) 的定义域由所有属于 \(g\) 定义域且 \(g(x)\) 属于 \(f\) 定义域的 \(x\) 组成。
For example, let \(f(x) = x^2\) and \(g(x) = x + 1\). Then \(f(g(x)) = (x+1)^2\), while \(g(f(x)) = x^2 + 1\). These are generally different, so composition is not commutative.
例如，设 \(f(x) = x^2\) 和 \(g(x) = x + 1\)。则 \(f(g(x)) = (x+1)^2\)，而 \(g(f(x)) = x^2 + 1\)。两者通常不同，因此复合不满足交换律。
When finding the domain of a composite, do not simply simplify the expression and then take the domain of the simplified version. Always consider the inner function’s output.
求复合函数定义域时，不要只化简表达式后取化简版本的定义域。始终要考虑内层函数的输出。
6. Inverse Functions | 反函数
The inverse of a function \(f\), denoted \(f^{-1}\), reverses the action of \(f\). Formally, for all \(x\) in the domain of \(f\), \(f^{-1}(f(x)) = x\), and for all \(y\) in the range of \(f\), \(f(f^{-1}(y)) = y\).
函数 \(f\) 的反函数记为 \(f^{-1}\)，它逆转 \(f\) 的作用。正式地，对于 \(f\) 定义域中的所有 \(x\)，有 \(f^{-1}(f(x)) = x\)；对于 \(f\) 值域中的所有 \(y\)，有 \(f(f^{-1}(y)) = y\)。
Not every function has an inverse. A function must be one-to-one (injective) to have an inverse function. The horizontal line test checks this: if any horizontal line intersects the graph more than once, the function is not one-to-one.
并非每个函数都有反函数。函数必须是一一对应（单射）才有反函数。水平线检验可判断：若任何水平线与图像相交多于一次，则该函数不是一一对应。
To find the inverse algebraically, swap \(x\) and \(y\) in the equation \(y = f(x)\), then solve for \(y\). The graph of \(f^{-1}\) is the reflection of the graph of \(f\) across the line \(y = x\).
代数求反函数的方法：在方程 \(y = f(x)\) 中交换 \(x\) 和 \(y\)，然后解出 \(y\)。\(f^{-1}\) 的图像是 \(f\) 的图像关于直线 \(y = x\) 的反射。
7. Even and Odd Functions | 奇偶性
A function \(f\) is even if \(f(-x) = f(x)\) for all \(x\) in its domain. Its graph is symmetric about the \(y\)-axis. Examples include \(f(x) = x^2\) and \(f(x) = \cos x\).
若对于定义域中所有 \(x\) 都有 \(f(-x) = f(x)\)，则函数 \(f\) 是偶函数。其图像关于 \(y\) 轴对称。例子包括 \(f(x) = x^2\) 和 \(f(x) = \cos x\)。
A function \(f\) is odd if \(f(-x) = -f(x)\) for all \(x\) in its domain. Its graph is symmetric about the origin. Examples include \(f(x) = x^3\) and \(f(x) = \sin x\).
若对于定义域中所有 \(x\) 都有 \(f(-x) = -f(x)\)，则函数 \(f\) 是奇函数。其图像关于原点对称。例子包括 \(f(x) = x^3\) 和 \(f(x) = \sin x\)。
Many functions are neither even nor odd. To test, compute \(f(-x)\) and compare with \(f(x)\) and \(-f(x)\). The product of two even or two odd functions is even; the product of an even and an odd function is odd.
许多函数既不是偶函数也不是奇函数。要检验，计算 \(f(-x)\) 并与 \(f(x)\) 和 \(-f(x)\) 比较。两个偶函数或两个奇函数的乘积是偶函数；偶函数与奇函数的乘积是奇函数。
8. Increasing and Decreasing Functions | 单调性
A function \(f\) is increasing on an interval if, for any \(x_1 < x_2\), \(f(x_1) < f(x_2)\). It is decreasing if \(f(x_1) > f(x_2)\). If the inequalities are non-strict, we say the function is non-decreasing or non-increasing.
若在某一区间内，对于任意 \(x_1 < x_2\)，都有 \(f(x_1) < f(x_2)\)，则函数 \(f\) 递增。若 \(f(x_1) > f(x_2)\)，则递减。若不等式为非严格的，则称函数非减或非增。
In IB problems, you may use a graph or a derivative to determine monotonicity. For a differentiable function, \(f'(x) > 0\) indicates increasing, and \(f'(x) < 0\) indicates decreasing, on intervals where the derivative does not change sign.
在 IB 题目中，你可以用图像或导数确定单调性。对于可导函数，在导数不改变符号的区间上，\(f'(x) > 0\) 表示递增，\(f'(x) < 0\) 表示递减。
Concavity is a related but distinct concept: it refers to the direction of bending, not the direction of motion. However, IB exam questions often ask you to describe both monotonicity and concavity from a given graph.
凹凸性是相关但不同的概念：它指的是弯曲的方向，而不是运动的方向。不过，IB 考试题经常要求你根据给定图像描述单调性和凹凸性。
9. Boundedness and Extrema | 有界性与极值
A function is bounded above if there is a number \(M\) such that \(f(x) \le M\) for all \(x\). It is bounded below if \(f(x) \ge m\) for all \(x\). If both hold, the function is bounded.
若存在一个数 \(M\) 使得对所有 \(x\) 都有 \(f(x) \le M\)，则函数有上界。若对所有 \(x\) 都有 \(f(x) \ge m\)，则有下界。若两者都成立，则函数有界。
The maximum and minimum values of a function on a given domain are called extrema. A local maximum occurs at a point where nearby values are all smaller; a global maximum occurs at a point where the point’s value is the largest in the entire domain.
函数在给定定义域上的最大值和最小值称为极值。局部最大值出现在附近所有值都更小的点；全局最大值出现在该点的值在整个定义域中最大的点。
For a quadratic function \(f(x) = ax^2 + bx + c\), if \(a > 0\), the vertex gives the minimum; if \(a < 0\), the vertex gives the maximum. The vertex \(x\)-coordinate is \(x = -\frac{b}{2a}\).
对于二次函数 \(f(x) = ax^2 + bx + c\)，若 \(a > 0\)，顶点给出最小值；若 \(a < 0\)，顶点给出最大值。顶点的 \(x\) 坐标为 \(x = -\frac{b}{2a}\)。
10. Periodicity and Symmetry | 周期性与对称性
A function \(f\) is periodic if there exists a positive constant \(p\) such that \(f(x + p) = f(x)\) for all \(x\). The smallest such \(p\) is called the fundamental period. Trigonometric functions like \(\sin x\) and \(\cos x\) have period \(2\pi\).
若存在正常数 \(p\) 使得对所有 \(x\) 都有 \(f(x + p) = f(x)\)，则函数 \(f\) 是周期函数。满足条件的最小正数 \(p\) 称为基本周期。三角函数如 \(\sin x\) 和 \(\cos x\) 的周期为 \(2\pi\)。
Symmetry is a visual property of graphs. Even symmetry about the \(y\)-axis and odd symmetry about the origin are special cases. More general transformations, such as reflection across a vertical line, can also appear in exam questions.
对称性是图像的视觉性质。关于 \(y\) 轴的偶对称和关于原点的奇对称是特殊情况。更一般的变换，例如关于某条垂直线的反射，也可能出现在考试题中。
When combining periodic functions, the period of the sum is the least common multiple of the individual periods, provided the periods are commensurable. For example, \(\sin x + \cos 2x\) has period \(2\pi\).
组合周期函数时，和函数的周期是各周期的最小公倍数，前提是各周期可比。例如，\(\sin x + \cos 2x\) 的周期为 \(2\pi\)。
11. Transformations of Graphs | 图像变换
Given the graph of \(y = f(x)\), several simple transformations can be applied:
给定 \(y = f(x)\) 的图像，可以应用几种简单变换：
- \(y = f(x) + k\) shifts the graph vertically by \(k\) units.
- \(y = f(x + h)\) shifts the graph horizontally by \(-h\) units.
- \(y = a f(x)\) stretches or compresses the graph vertically by a factor \(a\).
- \(y = f(bx)\) stretches or compresses the graph horizontally by a factor \(1/b\).
- \(y = -f(x)\) reflects the graph across the \(x\)-axis.
- \(y = f(-x)\) reflects the graph across the \(y\)-axis.
\(y = f(x) + k\) 将图像垂直平移 \(k\) 个单位。
\(y = f(x + h)\) 将图像水平平移 \(-h\) 个单位。
\(y = a f(x)\) 将图像垂直拉伸或压缩 \(a\) 倍。
\(y = f(bx)\) 将图像水平拉伸或压缩 \(1/b\) 倍。
\(y = -f(x)\) 将图像关于 \(x\) 轴反射。
\(y = f(-x)\) 将图像关于 \(y\) 轴反射。
Order matters when multiple transformations are applied. A common IB question asks for the image of a specific point, so track coordinates step by step.
应用多个变换时，顺序很重要。常见的 IB 题目要求某个特定点的像，因此要逐步追踪坐标。
12. Summary of Key Properties | 关键性质总结
To master functions, you should be able to quickly identify domain and range, determine whether a function is one-to-one, find inverses and composites, and classify symmetry and monotonic behaviour. These skills are heavily tested in both Paper 1 and Paper 2.
要掌握函数内容，你应该能够快速确定定义域和值域，判断函数是否一一对应，求反函数和复合函数，并分类对称性和单调行为。这些技能在卷一和卷二中都大量考查。
| Property | 性质 | Test / Meaning | 检验 / 含义 |
| Function | 函数 | Each input has exactly one output | 每个输入恰好一个输出 |
| One-to-one | 一一对应 | Horizontal line test | 水平线检验 |
| Even | 偶函数 | \(f(-x) = f(x)\) |
| Odd | 奇函数 | \(f(-x) = -f(x)\) |
| Periodic | 周期函数 | \(f(x+p) = f(x)\) |
Practice with past IB questions is essential. Start by sketching functions manually, then check with a graphing calculator. The more comfortable you are with the language of functions, the easier higher-level topics become.
练习 IB 历年真题至关重要。先手动画出函数图像，再用图形计算器验证。你对函数语言越熟悉，高阶主题就越容易掌握。
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