🔄 反函数：把机器倒着开🔄 Inverse Functions: running the machine backwards
上一片把两台机器串成一条流水线：先乘 2 再加 3。这一片反过来问：手里拿着最后的成品 13，能倒推出最初的原料吗？能——这就是反函数。Last leaf chained two machines into one line: multiply by 2, then add 3. This leaf flips the question: holding the final product, 13, can we recover the original input? Yes — that's the inverse function.
第 1 步：机器倒着开Step 1: Run the machine backwards
先看小算盘机器人的机器铺里有什么。First, a look inside the little abacus robot's machine shop.
正着开：5 进去，先 ×2 变 10，再 +3 出 13。倒着开：从 13 出发，先 −3 得 10，再 ÷2 得 5。顺序倒过来、每一步也反着做，就回到了原料。把正向机器记作 f(x) = 2x + 3，倒着开那台就叫 f⁻¹，写成 f⁻¹(13) = 5。Forward: 5 goes in, ×2 makes 10, +3 gives 13. Backwards: start from 13, −3 gives 10, ÷2 gives 5. Reverse the order and undo every step, and you're back to the input. Call the forward machine f(x) = 2x + 3; the one that runs backwards is f⁻¹, so f⁻¹(13) = 5.
倒着开的机器，它的图像长什么样？What does the backwards machine look like as a graph?
第 2 步：沿 y = x 折过去Step 2: Fold across y = x
把 f 的图像沿直线 y = x 像折纸一样翻过去，落点正好是 f⁻¹ 的图像。道理很简单：f 上的点 (a, b) 表示“输入 a、输出 b”；翻折后变成 (b, a)——输入输出刚好对调。所以两条曲线关于 y = x 镜像对称，一个的每个点，都是另一个的“倒过来”。Fold the graph of f across the line y = x like a sheet of paper, and it lands exactly on the graph of f⁻¹. Why? A point (a, b) on f means “input a, output b”; after folding it becomes (b, a) — input and output swapped. The two curves are mirror images across y = x: every point of one is the other one flipped.
那是不是每台机器都能倒着开？So can every machine be run backwards?
不一定。平方机器把 3 和 −3 都变成 9，倒着开就分不清原数是哪一个。限定只收正数（x ≥ 0）之后，倒挡装上了：开方。√9 = 3、√49 = 7——开方就是平方的反函数。想能倒着开，秘诀只有一条：一个输出，只能对应一个输入。Not always. The squaring machine turns both 3 and −3 into 9, so running backwards can't tell which one went in. Limit it to positive inputs (x ≥ 0) and the reverse gear fits: the square root. √9 = 3, √49 = 7 — the square root is the inverse of squaring. The secret to running backwards: one output must come from exactly one input.
🎮 你来倒着开机器（30 秒）🎮 Your turn: run the machine backwards (30 seconds)
现在你知道机器的输出，拖动滑块把输入还原出来。4 台机器，全部倒着开成功就通关。Now you know the machine's output — drag the slider to recover the input. Get all 4 machines running backwards to win.
一句话记住它：反函数 f⁻¹ 就是把机器倒着开——把输出还原成输入；图像上，它与 f 关于直线 y = x 镜像对称。Remember it in one line: the inverse f⁻¹ runs the machine backwards — it turns outputs back into inputs; on a graph, it mirrors f across the line y = x.
f 把 a 变成 b，f⁻¹ 就把 b 变回 a：输入输出对调If f turns a into b, then f⁻¹ turns b back into a — inputs and outputs swap倒着开 = 顺序倒过来 + 每一步都反着做（−3 再 ÷2）Backwards = reverse the order and undo each step (−3, then ÷2)开方是平方的反函数（限定 x ≥ 0）；一个输出只能对应一个输入The square root inverts squaring (with x ≥ 0); one output must map back to one input
内容参考 OpenStax 等公开教材，多来源核对 · AI 生成、人工审核 · 发现错误欢迎指正，帮这片叶子长得更好。Based on OpenStax and other open textbooks, cross-checked across sources · AI-generated, human-reviewed · Spotted a mistake? Tell us — help this leaf grow.
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