📚 Functions: Concepts and Basic Properties | 函数概念与基本性质
A function is one of the most fundamental ideas in A-Level Mathematics. It describes a precise relationship between two sets: each input is assigned exactly one output. Understanding the definition, notation, domain, range, and transformations of functions is essential for solving equations, sketching graphs, and modelling real-world situations.
函数是A-Level数学中最基本的概念之一。它描述了两个集合之间精确的对应关系：每一个输入值都恰好对应一个输出值。理解函数的定义、记号、定义域、值域以及图像变换，对于求解方程、绘制图像和建立现实模型都至关重要。
1. Definition of a Function | 函数的定义
In formal mathematics, a function \(f\) from a set \(X\) to a set \(Y\) is a rule that assigns to every element \(x\) in \(X\) exactly one element \(y\) in \(Y\). We write \(y = f(x)\). The key word is “exactly one”: a function cannot give two different outputs for the same input.
在正式数学中，从集合\(X\)到集合\(Y\)的函数\(f\)是一个规则，它将\(X\)中的每一个元素\(x\)恰好对应到\(Y\)中的一个元素\(y\)，记作\(y = f(x)\)。关键词是“恰好一个”：同一个输入值不能产生两个不同的输出值。
For example, \(f(x) = x^2\) is a function because every real number \(x\) gives exactly one square \(x^2\). However, the equation \(y^2 = x\) does not define a function of \(x\) for \(x > 0\), because a single \(x\) such as \(4\) gives two possible \(y\) values: \(2\) and \(-2\).
例如，\(f(x) = x^2\)是一个函数，因为每个实数\(x\)都恰好对应一个平方值\(x^2\)。然而，方程\(y^2 = x\)在\(x > 0\)时并不把\(y\)定义为\(x\)的函数，因为一个\(x\)值如\(4\)会产生两个可能的\(y\)值：\(2\)和\(-2\)。
There are several ways to represent a function: by a formula, a table, a graph, or a mapping diagram. Regardless of the form, the defining property remains the same: each input has only one output.
函数的表示方式有多种：公式、表格、图像或映射图。无论采用哪种形式，其本质属性不变：每个输入值只能有一个输出值。
2. Domain and Range | 定义域与值域
The domain of a function is the complete set of possible input values \(x\). The range is the complete set of possible output values \(f(x)\). In Edexcel A-Level questions, you must be careful to identify any restrictions on the domain, such as denominators cannot be zero and expressions under a square root must be non-negative.
函数的定义域是所有可能的输入值\(x\)构成的集合；值域是所有可能的输出值\(f(x)\)构成的集合。在Edexcel A-Level题目中，必须注意识别定义域上的限制，例如分母不能为零、根号下的表达式必须非负等。
For example, the function \(f(x) = \frac{1}{x-2}\) has domain \(x \in \mathbb{R}, x \neq 2\). Its range is all real numbers except \(0\), because the reciprocal of any non-zero number can be any non-zero number. Similarly, \(g(x) = \sqrt{3-x}\) has domain \(x \leq 3\) and range \(y \geq 0\).
例如，函数\(f(x) = \frac{1}{x-2}\)的定义域为\(x \in \mathbb{R}, x \neq 2\)。它的值域是除\(0\)以外的所有实数，因为任何非零数的倒数可以是任意非零数。类似地，\(g(x) = \sqrt{3-x}\)的定义域为\(x \leq 3\)，值域为\(y \geq 0\)。
When finding the range of a quadratic function, complete the square or use the vertex. For \(f(x) = x^2 – 4x + 7\), we rewrite it as \(f(x) = (x-2)^2 + 3\), so the range is \(f(x) \geq 3\).
求二次函数的值域时，可以通过配方法或利用顶点。对于\(f(x) = x^2 – 4x + 7\)，改写为\(f(x) = (x-2)^2 + 3\)，因此值域为\(f(x) \geq 3\)。
3. Function Notation and the Vertical Line Test | 函数记号与垂直线检验
Function notation \(f(x)\) allows us to evaluate expressions algebraically. For instance, if \(f(x) = 2x + 1\), then \(f(3) = 7\) and \(f(a+1) = 2(a+1) + 1 = 2a + 3\). The notation also helps us write composite functions clearly.
函数记号\(f(x)\)使我们能够进行代数运算。例如，若\(f(x) = 2x + 1\)，则\(f(3) = 7\)，且\(f(a+1) = 2(a+1) + 1 = 2a + 3\)。这一记号也使我们能够清晰地写出复合函数。
Graphically, a curve represents a function if and only if every vertical line intersects the curve at most once. This is called the vertical line test. A circle such as \(x^2 + y^2 = 1\) fails this test because a vertical line \(x = 0.5\) cuts the circle in two points.
在图像上，一条曲线表示一个函数当且仅当每一条垂直线与曲线的交点至多一个。这称为垂直线检验。像圆\(x^2 + y^2 = 1\)这样的曲线不满足该检验，因为垂直线\(x = 0.5\)与圆有两个交点。
You should also be familiar with the notation \(f: x \mapsto …\), which is common in Edexcel papers. For example, \(f: x \mapsto 5x – 3\) is the same as \(f(x) = 5x – 3\).
你还应熟悉记号\(f: x \mapsto …\)，这在Edexcel试卷中很常见。例如，\(f: x \mapsto 5x – 3\)与\(f(x) = 5x – 3\)相同。
4. Composite Functions | 复合函数
A composite function is formed when the output of one function becomes the input of another. If \(f(x)\) and \(g(x)\) are functions, then the composite function \(fg(x)\) (often written \(f \circ g(x)\)) means \(f(g(x))\). The order is extremely important: \(fg(x)\) is not generally equal to \(gf(x)\).
复合函数是将一个函数的输出作为另一个函数的输入所构成的函数。若\(f(x)\)和\(g(x)\)是函数，复合函数\(fg(x)\)（常写作\(f \circ g(x)\)）表示\(f(g(x))\)。运算顺序非常重要：\(fg(x)\)通常不等于\(gf(x)\)。
For example, let \(f(x) = 2x + 3\) and \(g(x) = x^2\). Then \(fg(x) = f(x^2) = 2x^2 + 3\), while \(gf(x) = g(2x+3) = (2x+3)^2\). These are different expressions.
例如，设\(f(x) = 2x + 3\)，\(g(x) = x^2\)。则\(fg(x) = f(x^2) = 2x^2 + 3\)，而\(gf(x) = g(2x+3) = (2x+3)^2\)。这两个表达式不同。
When finding the domain of a composite function, you must consider the domain of the inner function and then the domain of the outer function after substitution. For example, if \(f(x) = \frac{1}{x-1}\) and \(g(x) = x + 2\), then \(fg(x) = \frac{1}{(x+2)-1} = \frac{1}{x+1}\), so \(x \neq -1\).
求复合函数的定义域时，需要考虑内层函数的定义域以及代入后外层函数的定义域。例如，若\(f(x) = \frac{1}{x-1}\)，\(g(x) = x + 2\)，则\(fg(x) = \frac{1}{(x+2)-1} = \frac{1}{x+1}\)，因此\(x \neq -1\)。
5. Inverse Functions | 反函数
The inverse function undoes the effect of the original function. If \(f(x)\) maps \(x\) to \(y\), then the inverse function \(f^{-1}(y)\) maps \(y\) back to \(x\). The domain of \(f^{-1}\) is the range of \(f\), and the range of \(f^{-1}\) is the domain of \(f\).
反函数是撤销原函数作用的函数。若\(f(x)\)把\(x\)映射到\(y\)，那么反函数\(f^{-1}(y)\)将\(y\)映射回\(x\)。反函数的定义域是原函数的值域，反函数的值域是原函数的定义域。
For \(f\) to have an inverse, it must be one-to-one (injective). A one-to-one function has the property that no two different inputs give the same output. Graphically, it passes the horizontal line test. For example, \(f(x) = x^2\) with domain \(\mathbb{R}\) is not one-to-one, but if we restrict the domain to \(x \geq 0\), it becomes one-to-one and its inverse is \(f^{-1}(x) = \sqrt{x}\).
函数\(f\)要有反函数，必须是一一对应的（单射）。一一对应的函数满足：没有两个不同的输入会产生相同的输出。从图像上看，它可以通过水平线检验。例如，定义在\(\mathbb{R}\)上的\(f(x) = x^2\)不是一一对应，但如果将定义域限制为\(x \geq 0\)，它就变成一一对应，其反函数为\(f^{-1}(x) = \sqrt{x}\)。
To find an inverse algebraically: replace \(f(x)\) with \(y\), swap \(x\) and \(y\), then solve for \(y\). For \(f(x) = \frac{2x+1}{x-3}\), we write \(x = \frac{2y+1}{y-3}\). Rearranging gives \(x(y-3) = 2y+1\), so \(y = \frac{3x+1}{x-2}\). Therefore \(f^{-1}(x) = \frac{3x+1}{x-2}\), with \(x \neq 2\).
代数上求反函数的步骤为：将\(f(x)\)写成\(y\)，交换\(x\)和\(y\)，然后解出\(y\)。对于\(f(x) = \frac{2x+1}{x-3}\)，令\(x = \frac{2y+1}{y-3}\)。整理得\(x(y-3) = 2y+1\)，所以\(y = \frac{3x+1}{x-2}\)。因此\(f^{-1}(x) = \frac{3x+1}{x-2}\)，其中\(x \neq 2\)。
6. Even and Odd Functions | 函数的奇偶性
An even function satisfies \(f(-x) = f(x)\) for all \(x\) in its domain. Its graph is symmetrical about the \(y\)-axis. Examples include \(f(x) = x^2\), \(f(x) = x^4\), and \(f(x) = \cos x\). An odd function satisfies \(f(-x) = -f(x)\) for all \(x\). Its graph has rotational symmetry of order 2 about the origin. Examples include \(f(x) = x^3\), \(f(x) = x\), and \(f(x) = \sin x\).
偶函数满足对定义域内所有\(x\)，有\(f(-x) = f(x)\)。其图像关于\(y\)轴对称。例子包括\(f(x) = x^2\)、\(f(x) = x^4\)和\(f(x) = \cos x\)。奇函数满足对定义域内所有\(x\)，有\(f(-x) = -f(x)\)。其图像关于原点具有二阶旋转对称性。例子包括\(f(x) = x^3\)、\(f(x) = x\)和\(f(x) = \sin x\)。
Many functions are neither even nor odd. For instance, \(f(x) = x^2 + x\) does not satisfy either condition. In Edexcel exams, you may be asked to determine whether a given function is even, odd, or neither. It is helpful to check \(f(-x)\) algebraically before graphing.
许多函数既不是偶函数也不是奇函数。例如，\(f(x) = x^2 + x\)既不满足偶函数条件也不满足奇函数条件。在Edexcel考试中，你可能会被要求判断给定函数是偶函数、奇函数还是两者都不是。先在代数上检查\(f(-x)\)再画图是有帮助的。
A useful property: the product of two even functions is even, the product of two odd functions is even, and the product of an even and an odd function is odd. These relationships can be used for quick verification.
一个有用的性质：两个偶函数的乘积是偶函数；两个奇函数的乘积是偶函数；一个偶函数与一个奇函数的乘积是奇函数。这些关系可用于快速验证。
7. Increasing and Decreasing Functions | 函数的单调性
A function is increasing on an interval if \(x_1 < x_2\) implies \(f(x_1) < f(x_2)\). It is decreasing if \(x_1 < x_2\) implies \(f(x_1) > f(x_2)\). A function that is either always increasing or always decreasing on its whole domain is called monotonic.
函数在某个区间上递增是指：若\(x_1 < x_2\)，则\(f(x_1) < f(x_2)\)。递减是指：若\(x_1 < x_2\)，则\(f(x_1) > f(x_2)\)。如果函数在整个定义域上始终递增或始终递减，则称为单调函数。
For a differentiable function \(f\), you can use the sign of the derivative \(f'(x)\): if \(f'(x) > 0\) for all \(x\) in an interval, the function is increasing there; if \(f'(x) < 0\), it is decreasing. For example, \(f(x) = x^3 - 3x\) has derivative \(f'(x) = 3x^2 - 3 = 3(x-1)(x+1)\). Thus it is increasing for \(x < -1\) and \(x > 1\), and decreasing for \(-1 < x < 1\).
对于可微函数\(f\)，可以用导数\(f'(x)\)的符号来判断：若在某一区间内\(f'(x) > 0\)，则函数在该区间递增；若\(f'(x) < 0\)，则递减。例如，\(f(x) = x^3 - 3x\)的导数为\(f'(x) = 3x^2 - 3 = 3(x-1)(x+1)\)。因此，在\(x < -1\)和\(x > 1\)上递增，在\(-1 < x < 1\)上递减。
Points where the derivative is zero are called stationary points. They can be local maxima, local minima, or points of inflection. The second derivative \(f”(x)\) helps distinguish these: \(f”(x) < 0\) indicates a local maximum, and \(f''(x) > 0\) indicates a local minimum.
导数为零的点称为驻点。它们可能是局部最大值、局部最小值或拐点。二阶导数\(f”(x)\)有助于区分这些情况：\(f”(x) < 0\)表示局部最大值，\(f''(x) > 0\)表示局部最小值。
8. Transformations of Graphs | 图像变换
The graph of a function can be transformed in four basic ways. The transformation \(y = f(x) + a\) shifts the graph vertically by \(a\) units. The transformation \(y = f(x + a)\) shifts horizontally by \(-a\) units. The transformation \(y = -f(x)\) reflects the graph in the \(x\)-axis, and \(y = f(-x)\) reflects it in the \(y\)-axis.
函数图像可以通过四种基本方式进行变换。\(y = f(x) + a\)将图像垂直平移\(a\)个单位。\(y = f(x + a)\)将图像水平平移\(-a\)个单位。\(y = -f(x)\)将图像关于\(x\)轴对称反射，\(y = f(-x)\)将图像关于\(y\)轴对称反射。
Scaling transformations include \(y = af(x)\), which stretches the graph vertically by a factor of \(a\), and \(y = f(ax)\), which compresses it horizontally by a factor of \(a\) (for \(a > 0\)). For example, starting from \(f(x) = x^2\), the function \(y = 2(x-1)^2 + 3\) is obtained by a vertical stretch factor 2, a horizontal shift right by 1, and a vertical shift up by 3.
缩放变换包括：\(y = af(x)\)将图像沿垂直方向拉伸\(a\)倍；\(y = f(ax)\)将图像沿水平方向压缩\(a\)倍（\(a > 0\)）。例如，从\(f(x) = x^2\)出发，函数\(y = 2(x-1)^2 + 3\)可通过垂直拉伸2倍、水平向右平移1个单位、垂直向上平移3个单位得到。
When multiple transformations are applied, the order matters. A common sequence is: horizontal shift, then vertical stretch, then reflection, then vertical shift. Always study the algebraic form carefully to determine the correct order.
当应用多个变换时，顺序很重要。常见顺序为：水平平移、垂直拉伸、反射、垂直平移。始终仔细研究代数形式，以确定正确的顺序。
9. One-to-One Functions and the Horizontal Line Test | 一一对应与水平线检验
A one-to-one function is a function where every element in the range corresponds to exactly one element in the domain. In other words, if \(x_1 \neq x_2\), then \(f(x_1) \neq f(x_2)\). The horizontal line test states that a function is one-to-one if every horizontal line intersects its graph at most once.
一一对应函数是指值域中每个元素恰好与定义域中一个元素对应的函数。换句话说，若\(x_1 \neq x_2\)，则\(f(x_1) \neq f(x_2)\)。水平线检验指出：如果每条水平线与图像的交点至多一个，则该函数是一一对应的。
This property is essential for the existence of an inverse function. For example, \(f(x) = e^x\) is one-to-one and its inverse is \(f^{-1}(x) = \ln x\). But \(f(x) = \sin x\) is not one-to-one over \(\mathbb{R}\); its domain must be restricted to \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\) to define the inverse function \(\sin^{-1} x\).
这一性质对于反函数的存在至关重要。例如，\(f(x) = e^x\)是一一对应的，其反函数为\(f^{-1}(x) = \ln x\)。但\(f(x) = \sin x\)在\(\mathbb{R}\)上不是一一对应；必须将定义域限制为\(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\)才能定义反函数\(\sin^{-1} x\)。
When asked to restrict a domain to make a function invertible, choose an interval that contains the vertex or the turning point, and ensure that the function is monotonic on that interval.
当题目要求限制定义域以使函数可逆时，应选择一个包含顶点或转折点的区间，并确保函数在该区间上是单调的。
10. Piecewise Functions | 分段函数
A piecewise function is defined by different rules for different parts of its domain. It is still a single function, provided each input falls into exactly one piece. For example:
分段函数在其定义域的不同部分使用不同的规则。只要每个输入值恰好属于某一个分支，它仍然是一个函数。例如：
f(x) = x² for x ≤ 1, and f(x) = 2x + 3 for x > 1
This function has a jump at \(x = 1\): the left-hand value is \(1^2 = 1\), while the right-hand value is \(2(1) + 3 = 5\). Such discontinuities are common in real-world situations like tax brackets or shipping costs.
该函数在\(x = 1\)处有一个跳跃：左侧值为\(1^2 = 1\)，右侧值为\(2(1) + 3 = 5\)。这种不连续在现实情境中很常见，例如税率等级或运费计算。
To sketch a piecewise function, draw each branch on its specified interval, and pay attention to whether the endpoint is included (use a filled dot) or excluded (use an open dot).
绘制分段函数的图像时，应在指定区间内画出每个分支，并注意端点是否包含在内（包含用实心点，不包含用空心点）。
11. The Modulus Function | 绝对值函数
The modulus function is defined by \(|x| = x\) when \(x \geq 0\), and \(|x| = -x\) when \(x < 0\). It measures the distance from zero on the number line. The graph of \(y = |f(x)|\) is obtained from the graph of \(y = f(x)\) by reflecting the part below the \(x\)-axis up into the positive region.
绝对值函数定义为：当\(x \geq 0\)时，\(|x| = x\)；当\(x < 0\)时，\(|x| = -x\)。它表示数轴上到零的距离。\(y = |f(x)|\)的图像可通过将\(y = f(x)\)图像中位于\(x\)轴下方的部分向上翻折得到。
For example, \(y = |x-2|\) has a V-shape with the vertex at \((2,0)\). Solving equations involving modulus requires considering separate cases. The equation \(|x-2| = 5\) means \(x-2 = 5\) or \(x-2 = -5\), giving \(x = 7\) or \(x = -3\).
例如，\(y = |x-2|\)的图像是V形，顶点在\((2,0)\)。求解含有绝对值的方程需要分情况讨论。方程\(|x-2| = 5\)意味着\(x-2 = 5\)或\(x-2 = -5\)，得到\(x = 7\)或\(x = -3\)。
Inequalities with modulus also require case work. For \(|2x – 1| < 3\), we write \(-3 < 2x - 1 < 3\), hence \(-1 < x < 2\). The graph-based interpretation is equally useful: the set of points where the curve lies below the horizontal line \(y = 3\).
含绝对值的不等式也需要分类讨论。对于\(|2x – 1| < 3\)，我们写为\(-3 < 2x - 1 < 3\)，因此\(-1 < x < 2\)。基于图像的理解同样有用：即曲线位于水平线\(y = 3\)下方的点集。
12. Summary and Exam Tips | 总结与考试要点
A strong command of function concepts is vital for success in Edexcel A-Level Mathematics. Always state the domain before finding the range. Check whether a function is one-to-one before finding its inverse. When working with composite functions, pay attention to the order. For transformations, apply horizontal shifts before stretching unless the equation indicates otherwise.
扎实掌握函数概念对于在Edexcel A-Level数学中取得成功至关重要。求值域前先写出定义域；求反函数前先检查函数是否一一对应；处理复合函数时注意运算顺序；对于图像变换，除非方程另有提示，否则先进行水平平移再进行拉伸。
In exams, show all intermediate steps, especially when solving modulus equations or determining the domain of a composite function. Use the vertical and horizontal line tests to classify graphs quickly. Practising past paper questions will help you recognise common traps, such as forgetting that \(f(-x)\) is not the same as \(-f(x)\).
在考试中，要展示所有中间步骤，尤其是在求解绝对值方程或确定复合函数定义域时。使用垂直线检验和水平线检验快速判断图像类型。通过练习历年真题，你会识别出常见陷阱，例如忘记\(f(-x)\)与\(-f(x)\)并不相同。
Finally, remember that functions are not just abstract objects. They model real-world relationships: exponential functions describe growth, trigonometric functions describe periodic motion, and quadratic functions describe projectile paths. Understanding their basic properties gives you the tools to analyse them effectively.
最后，请记住函数不仅仅是抽象对象。它们可以用来建立现实世界中的关系模型：指数函数描述增长，三角函数描述周期运动，二次函数描述抛物体路径。理解它们的基本性质，能够为你提供有效分析的工具。
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