Rearranging Functions | 函数的重排

📚 Rearranging Functions | 函数的重排

Rearranging functions is a fundamental skill in IB Mathematics, involving techniques to manipulate functions into different forms. This includes finding inverse functions, composing functions, applying transformations to graphs, and using these rearrangements to solve equations. Mastering these concepts allows you to understand deep relationships between algebraic expressions and their graphical representations, and is essential for success in both Analysis and Approaches and Applications and Interpretation courses.

函数的重排是 IB 数学中的一项基本技能，涉及将函数转化为不同形式的各种技巧。这包括求反函数、构建复合函数、对图像应用变换以及利用这些重排来解方程。掌握这些概念能让你理解代数表达式与其图像表示之间的深层关系，对于在分析与方法和应用与解释两门课程中取得成功至关重要。

1. What Is Rearranging a Function? | 什么是函数的重排？

Rearranging a function refers to any process that changes the way a function is expressed without altering the underlying input-output relationship. This can mean solving for the input in terms of the output (finding the inverse), combining multiple functions into a single expression (composition), or shifting, stretching, and reflecting its graph. Each rearrangement reveals different properties, such as symmetry, asymptotic behaviour, or the solution to an equation.

函数的重排指在不改变其根本输入输出关系的前提下，改变函数表达方式的任何过程。这可以意味着解出用输出表示的输入（求反函数）、将多个函数组合成一个表达式（复合），或者平移、伸缩和反射其图像。每一种重排都会揭示不同的性质，例如对称性、渐近行为或方程的解。

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2. Inverse Functions: Definition and Notation | 反函数：定义与记号

Given a function f, its inverse, denoted f⁻¹, reverses the mapping: if f(a) = b, then f⁻¹ a function, f must be one-to-one (injective) on its domain. The defining equations are f(f⁻¹(x)) = x for all x in the domain of f⁻¹, and f⁻¹(f(x)) = x for all x in the domain of f. Note that f⁻¹ does not mean the reciprocal; it is functional notation.

给定函数 f，其反函数记作 f⁻¹，它会逆转映射关系：若 f(a) = b，则 f⁻¹(b) = a。要使反函数存在，f 在其定义域上必须是一一映射（单射）。定义方程是：对 f⁻¹ 定义域中的所有 x，有 f(f⁻¹(x)) = x；对 f 定义域中的所有 x，有 f⁻¹(f(x)) = x。请注意，f⁻¹ 并非表示倒数，它是一种函数记号。

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3. Finding the Inverse Function Algebraically | 代数求反函数

taining x = f(y); (3) solve this equation for y in terms of x; (4) rewrite the result as f⁻¹(x). For example, if f(x) = 2x + 3, then y = 2x + 3 → x = 2y + 3 → y = (x − 3)/2, so f⁻¹ ons, you must often restrict the domain to ensure one-to-oneness before finding the inverse.

要代数地求反函数，请遵循以下步骤：（1）写出 y = f(x)；（2）交换 x 和 y，得到 x = f(y)；（3）解出用 x 表示的 y；（4）将结果改写为 f⁻¹(x)。例如，若 f(x) = 2x + 3，则 y = 2x + 3 → x = 2y + 3 → y = (x − 3)/2，因此 f⁻¹(x) = (x − 3)/2。对于非线性函数，你通常需要先限制定义域以确保一一对应，然后再求反函数。

Always verify your inverse by checking that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. For instance, with f(x) = 2x + 3: f(f⁻¹(x)) = 2((x−3)/2)+3 = x, confirming correctness.

务必通过检查 f(f⁻¹(x)) = x 和 f⁻¹(f(x)) = x 来验证反函数。例如，对于 f(x) = 2x + 3：f(f⁻¹(x)) = 2((x−3)/2)+3 = x，确认无误。

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4. Domain and Range of Inverse Functions | 反函数的定义域与值域

The domain of f⁻¹ is exactly the range of f, and the range of f⁻¹ is the domain of f. This relationship provides a quick way to determine the inverse' omain and range. If f is defined on a restricted domain to make it one-to-one, the inverse's range is limited accordingly. For example, f(x) = x² with domain x ≥ 0 has range y ≥ 0; its inverse f⁻¹(x) = √x has domain x ≥ 0 and range y ≥ 0.

f⁻¹ 的定义域恰好是 f 的值域，而 f⁻¹ 的值域则是 f 的定义域。这一关系为确定反函数的定义域和值域提供了快捷方法。若 f 被定义在受限的定义域上以使其成为一一映射，则反函数的值域也会相应受限。例如，f(x) = x² 在 x ≥ 0 上的值域为 y ≥ 0；其反函数 f⁻¹(x) = √x 的定义域为 x ≥ 0，值域为 y ≥ 0。

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5. Graphical Relationship Between a Function and Its Inverse | 函数与反函数的图像关系

The graphs of f and f⁻¹ x. If a point (a, b) lies on the graph of f, then the point (b, a) lies on the graph of f⁻¹. This symmetry is a powerful visual tool: you can sketch the inverse by reflecting key points or the entire curve. It also helps solve equations graphically: the intersection of f and f⁻¹ lies on y = x (provided the functions are increasing).

f 与 f⁻¹ 的图像关于直线 y = x 对称。若点 (a, b) 在 f 的图像上，则点 (b, a) 在 f⁻¹ 的图像上。这种对称性是一个强大的视觉工具：你可以通过反射关键点或整个曲线来勾勒反函数。它还有助于以图像方式解方程：f 与 f⁻¹ 的交点位于 y = x 上（前提是函数为递增函数）。

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6. Composite Functions | 复合函数

The composition of two functions f and g, written (f ∘ x)) is generally not equal to g(f(x)). For example, if f(x) = 2x and g(x) = x + 3, then (f ∘ g)(x) = 2(x + 3) = 2x + 6, while (g ∘ different. Composite functions allow you to build complex relationships from simpler ones.

两个函数 f 和 g 的复合写作 (f ∘ g)(x) = f(g(x))，表示先作用 g，再作用 f。顺序很重要，因为 f(g(x)) 通常不等于 g(f(x))。例如，若 f(x) = 2x，g(x) = x + 3，则 (f ∘ g)(x) = 2(x + 3) = 2x + 6，而 (g ∘ f)(x) = (2x) + 3 = 2x + 3，两者不同。复合函数使你可以从简单的函数构建复杂的关系。

The symbol ∘ is read 'composed with'. You can compose more than two functions, e.g., (f ∘ g ∘ h)(x) = f(g(h(x))). Always work from the innermost bracket outward.

符号 ∘ 读作“复合于”。你可以对两个以上的函数进行复合，例如 (f ∘ g ∘ h)(x) = f(g(h(x)))。始终从最内层括号向外计算。

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7. Domain of Composite Functions | 复合函数的定义域

To find the domain of (f ∘ need two conditions: x must be in the domain of g, and g(x) must be in the domain of f. It's helpful to first determine the 'inner' g), then exclude any x-values for which g(x) is not allowed in f. For instance, if f(x) = √x (domain x ≥ 0) and g(x) = x − 2 (domain all real numbers), then (f ∘ g)(x) = √x − 2 requires x − 2 ≥ 0, so the domain is x ≥ 2.

要找出 (f ∘ g)(x) 的定义域，你需要满足两个条件：x 必须在 g 的定义域内，并且 g(x) 必须在 f 的定义域内。这有助于首先确定“内部”定义域（g 的定义域），然后排除那些使得 g(x) 不在 f 中允许的 x 值。例如，若 f(x) = √x（定义域 x ≥ 0）且 g(x) = x − 2（定义域为所有实数），则 (f ∘ g)(x) = √x − 2 要求 x − 2 ≥ 0，因此定义域为 x ≥ 2。

Restrictions can arise from denominators, square roots, logarithms, or other undefined operations. Always express the final domain in set notation or interval notation as required by the IB markscheme.

限制可能来自分母、平方根、对数或其他未定义的运算。始终按 IB 评分方案的要求，将最终定义域用集合记号或区间记号表示。

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8. Transformations of Functions: Translations | 函数的变换：平移

e graph shifts to the right; if h < 0, it shifts to the left. Be careful with the sign inside the bracket: f(x − 2) is a shift 2 units right, not left.

平移会在不改变形状的情况下移动函数的图像。垂直平移由 y = f(x) + k 给出：若 k > 0，图像上移；若 k < 0，则下移。水平平移由 y = f(x − h) 给出：若 h > 0，图像向右平移；若 h < 0，则向左平移。注意括号内的符号：f(x − 2) 是向右平移 2 个单位，而非向左。

For example, starting from f(x) = x², the graph of f(x) + 3 is a parabola shifted up 3 units, and f(x + 4) shifts the parabola left 4 units. Combined translations can be written as f(x − h) + k.

例如，从 f(x) = x² 出发，f(x) + 3 的图像是向上平移 3 个单位的抛物线，而 f(x + 4) 则将抛物线向左平移 4 个单位。组合平移可写作 f(x − h) + k。

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9. Transformations: Stretches and Reflections | 伸缩与反射

nge the scale of the graph. A vertical stretch (or compression) is given by y = a f(x): if |a| > 1, the graph stretches vertically away from the x-axis; if 0 < |a| < 1, it compresses vertically. A horizontal stretch is given by y = f(bx): if |b| > 1, the graph compresses horizontally towards the y-axis; if 0 < |b| < 1, it stretches horizontally. Note the 'reversed' intuition for horizontal scaling.

伸缩会改变图像的比例。垂直伸缩（或压缩）由 y = a f(x) 给出：若 |a| > 1，图像沿垂直方向远离 x 轴拉伸；若 0 < |a| < 1，则沿垂直方向压缩。水平伸缩由 y = f(bx) 给出：若 |b| > 1，图像朝 y 轴水平压缩；若 0 < |b| < 1，则水平拉伸。请注意水平缩放中“相反”的直觉。

h across the x-axis, while y = f(−x) reflects it across the y-axis. For example, if f(x) = √x, then y = −√x is the upper half-parabola flipped downward, and y = √−x reflects it to the left side (domain x ≤ 0).

反射会将图像翻转到轴的另一侧。y = −f(x) 将图像关于 x 轴反射，而 y = f(−x) 将其关于 y 轴反射。例如，若 f(x) = √x，则 y = −√x 是将上半支抛物线翻转向下，y = √−x 则将其反射到左侧（定义域 x ≤ 0）。

Transformation	Equation	Effect on graph
Vertical translation	y = f(x) + k	Shifts up/down by k
Horizontal translation	y = f(x − h)	Shifts right/left by h
Vertical stretch	y = a f(x)	Stretch by factor a vertically
Horizontal stretch	y = f(bx)	Stretch by factor 1/b horizontally
Reflection in x-axis	y = −f(x)	Flips over x-axis
Reflection in y-axis	y = f(−x)	Flips over y-axis

The table above summarises the six basic transformations. Memorising their effects helps you quickly sketch transformed graphs without plotting many points.

上表总结了六种基本变换。记住它们的效果有助于你快速描绘变换后的图像，而无需逐点描点。

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10. Combining Transformations | 组合变换

ons inside the function argument (on x) are applied first, right to left, relative to x. For example, y = 2 f(3(x + 1)) − 4: shift left 1, compress horizontally by factor 1/3 (since b=3), stretch vertically by factor 2, then shift down 4.

当应用多个变换时，顺序很重要。标准表达式 y = a f(b(x − h)) + k 描述了一个序列：先水平平移 h，再以 1/b 为因子水平伸缩，然后以 a 为因子垂直伸缩，最后垂直平移 k。函数参数内部（对 x）的操作是相对于 x 从右向左先应用。例如，y = 2 f(3(x + 1)) − 4：向左平移 1，以因子 1/3 水平压缩（因为 b=3），以因子 2 垂直拉伸，再向下平移 4。

= f(2x + 6) should be rewritten as f(2(x + 3)) to see the horizontal translation clearly.

反射等变换可以通过使 a 或 b 取负值来纳入。总是将表达式改写为清晰识别 f(b(x − h)) 的形式。例如，y = f(2x + 6) 应改写为 f(2(x + 3))，以便清楚看到水平平移。

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11. Rearranging Functions to Solve Equations | 重排函数以解方程

can express an equation as f(x) = k where f is invertible, you can immediately write x = f⁻¹ r example, solve exp(2x) = 5: let f(x) = exp(x), g(x) = 2x, then f(g(x)) = exp(2x). But it' Here you rearranged the exponential function into a logarithmic one, which is its inverse.

函数的重排是解方程的强大工具。如果你能将方程表示为 f(x) = k，其中 f 可逆，那么你可以立即写出 x = f⁻¹(k)。复合函数也会出现在方程中：解 f(g(x)) = a 可能需要先解出 g(x)，然后使用反函数。例如，解 exp(2x) = 5：令 f(x) = exp(x)，g(x) = 2x，则 f(g(x)) = exp(2x)。但更简单的方法是取自然对数：2x = ln(5)，x = 0.5 ln(5)。这里你将指数函数重排成了其反函数——对数函数。

d roots. Transformations can turn a complicated function into a simpler one: e.g., solving |x − 2| = 3 can be seen as shifting the absolute value function and finding where it equals 3.

图像的重排也有用：方程 f(x) = g(x) 的解对应图像交点的 x 坐标。你可以将方程重排为 0 = f(x) − g(x) 并求根。变换可以把复杂的函数变成简单的函数：例如，解 |x − 2| = 3 可以看作是将绝对值函数平移并找出它何时等于 3。

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12. Common Mistakes and Tips | 常见错误与提示

Mistake 1: Confusing inverse notation f⁻¹(x) with reciprocal 1/f(x). The inverse reverses the mapping, while the reciprocal is a completely different function.

错误 1：混淆反函数记号 f⁻¹(x) 与倒数 1/f(x)。反函数反转映射关系，而倒数则是完全不同的函数。

Mistake 2: Forgetting to restrict the domain when finding the inverse of a non-one-to-one function like a quadratic. Always state the domain restriction after inversion.

错误 2：在求非一一对应函数（如二次函数）的反函数时忘记限制定义域。在反函数后务必说明定义域限制。

Mistake 3: Applying composite functions in the wrong order. Remember: (f ∘ g)(x) = f(g(x)), apply g first.

错误 3：以错误的顺序应用复合函数。请记住：(f ∘ g)(x) = f(g(x))，先应用 g。

Mistake 4: = f(2x) compresses, not stretches; y = f(x + 3) shifts left, not right. Think in terms of 'what must x be to achieve the same output as before'.

错误 4：错误处理水平变换。y = f(2x) 的图像是压缩，而不是拉伸；y = f(x + 3) 向左平移，而非向右。要从“x 应取何值才能得到与之前相同的输出”的角度来思考。

Tip: Always check your answer by substituting a test point. For inverses, use the identity property. For transformations, pick a key point on the original graph and apply the transformation to see if it matches the new graph.

提示：始终通过代入测试点来检查答案。对于反函数，利用恒等性检查；对于变换，选取原图像上的一个关键点，应用变换后看其是否与新图像一致。

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