Functions and graphs | 函数与图像

📚 Functions and graphs | 函数与图像

In Edexcel A Level Mathematics, the topic of functions and graphs underpins much of pure mathematics. A function describes a precise relationship between inputs and outputs, while its graph gives a visual way to understand domain, range, transformations and solutions to equations.

在爱德思 A Level 数学中，函数与图像是纯数学许多内容的基础。函数描述输入与输出之间精确的关系，而图像则提供了理解定义域、值域、变换以及方程解的直观方式。

1. What is a function? | 什么是函数？

A function maps every input value from its domain to exactly one output value. This uniqueness is essential: if one input gives two different outputs, the rule is not a function.

函数将定义域中的每个输入值都唯一地映射到一个输出值。这种唯一性至关重要：如果一个输入对应两个不同输出，那么该规则就不是函数。

The vertical line test checks whether a graph represents a function. If any vertical line cuts the graph more than once, then one x-value has several y-values, so the graph is not a function.

垂直线检验用于判断一个图像是否表示函数。如果任意一条垂直线与图像相交超过一次，那么一个 x 值对应多个 y 值，因此该图像不是函数。

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2. Domain and range | 定义域与值域

The domain is the set of all possible input values for which the function is defined, while the range is the set of all possible output values produced by those inputs.

定义域是使函数有定义的所有可能输入值的集合，而值域是由这些输入产生的所有可能输出值的集合。

For example, if f(x) = √(x − 2), the expression under the square root must be non-negative, so x − 2 ≥ 0.

例如，如果 f(x) = √(x − 2)，根号下的表达式必须非负，因此 x − 2 ≥ 0。

f(x) = √(x − 2), domain x ≥ 2, range f(x) ≥ 0

This gives the domain x ≥ 2 and, since a square root is non-negative, the range is f(x) ≥ 0.

由此得到定义域 x ≥ 2；由于平方根非负，值域为 f(x) ≥ 0。

For a quadratic such as g(x) = x², the domain is all real numbers but the range is g(x) ≥ 0 because squares are never negative.

对于二次函数 g(x) = x²，定义域是所有实数，但值域是 g(x) ≥ 0，因为平方数永远不会为负。

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3. Function notation and evaluation | 函数符号与求值

Function notation f(x) means ‘f of x’ and is used to show the output of a function f for an input x. To evaluate f(3), replace every x in the rule by 3.

函数符号 f(x) 表示 ‘f of x’，用来表示函数 f 对输入 x 的输出。要求 f(3)，只需将规则中的每个 x 替换为 3。

If f(x) = 3x² − 5, then f(2) = 3(2)² − 5 = 7. This process is called substitution.

如果 f(x) = 3x² − 5，那么 f(2) = 3(2)² − 5 = 7。这个过程称为代入。

f(2) = 3(2)² − 5 = 7

Function notation also lets us use a variable input, such as f(a + h), which is useful when studying rates of change and gradients.

函数符号还允许我们使用变量输入，例如 f(a + h)，这在研究变化率和斜率时非常有用。

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4. Composite functions | 复合函数

A composite function combines two functions by using the output of one as the input of the other. In Edexcel notation, gf(x) means g(f(x)), so apply f first and then g. Similarly, fg(x) means f(g(x)).

复合函数将两个函数组合起来，将一个函数的输出作为另一个函数的输入。在爱德思记号中，gf(x) 表示 g(f(x))，即先作用 f，再作用 g。类似地，fg(x) 表示 f(g(x))。

For f(x) = 2x + 1 and g(x) = x², we obtain gf(x) = (2x + 1)² but fg(x) = 2x² + 1.

对于 f(x) = 2x + 1 和 g(x) = x²，我们得到 gf(x) = (2x + 1)²，而 fg(x) = 2x² + 1。

gf(x) = (2x + 1)², fg(x) = 2x² + 1

This shows that composite functions are usually not commutative: gf(x) is not the same as fg(x) in general.

这表明复合函数通常不满足交换律：一般来说 gf(x) 不等于 fg(x)。

The domain of a composite function is restricted: every x must belong to the domain of the inner function, and the resulting output must belong to the domain of the outer function.

复合函数的定义域有限制：每个 x 必须属于内层函数的定义域，并且得到的输出必须属于外层函数的定义域。

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5. Inverse functions | 反函数

The inverse function f⁻¹ reverses the effect of f. If f maps a to b, then f⁻¹ maps b back to a. For an inverse to exist, f must be one-to-one.

反函数 f⁻¹ 逆转函数 f 的作用。如果 f 将 a 映射到 b，那么 f⁻¹ 将 b 映射回 a。反函数存在的前提是 f 必须是一一对应的。

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