🚧 定义域与值域：哪些输入合法，哪些输出能出🚧 Domain & Range: which inputs may enter, which outputs can leave

上一片我们造好了函数机器，规矩是“一个输入一个输出”。可今天工厂门口排起了长队：有的数字被拦在门外，有的数字进去了却等不到结果。这一片管两件事——哪些 x 能进，哪些 y 能出。 Last leaf we built our function machines, with the rule “one input, one output”. But today a queue forms at the factory gate: some numbers are turned away, and some that walk in never produce a result. This leaf handles two things — which x may enter, and which y can come out.

第 1 步：门口的门槛Step 1: The threshold at the gate

先看一台带根号的机器：y = √x。Start with a machine that carries a square root: y = √x.

根号门：负数被拦下 — 根号在实数里是“求一个数，让它自乘等于 x”。负数自乘永远是正的（或零），所以 √(-4) 算不出实数。质检员只好在门槛竖起牌子：负数请回。所有被允许进门的 x 凑在一起，就叫这台机器的**定义域**（domain）。 A square root asks “which number times itself gives x?” A negative number times itself is always positive (or zero), so √(-4) has no real answer. The inspector puts up a sign: negatives, turn back. All the x values allowed in make up the **domain** of the machine.

换成带除法的机器，门槛又换了一把尺。Swap to a machine with division, and the threshold changes.

第 2 步：除数不能是 0Step 2: You can't divide by zero

数轴：x ≠ 0 — 在 y = 1/x 里，x 一旦等于 0，机器就撞上 1 ÷ 0——没有意义，门卫把 0 拦下。把允许范围涂在数轴上：√x 是 x ≥ 0 的一整片绿；1/x 是除了 0 以外全绿。最常见的两把门卫尺：**分母不为零、根号内非负**。 In y = 1/x, the moment x = 0 the machine faces 1 ÷ 0 — meaningless, so the gatekeeper stops 0. Paint the allowed stretch on the number line: for √x it is one long green run x ≥ 0; for 1/x everything is green except 0. The two most common gate rules: **denominator ≠ 0, inside a root ≥ 0**.

进了门的数字，出来时也有一道讲究。And after walking in, numbers face one more rule on the way out.

第 3 步：输出池里有几颗球Step 3: How many balls in the output pool

输入池到输出池：y = x² — 看机器 y = x²：输入随便挑（定义域是所有实数），但任何实数平方后都不是负数——出口的池子里只漂着 y ≥ 0 的球。所有可能输出的集合叫值域（range）。一句话：定义域管入口，值域管出口。 Take y = x²: feed it anything (the domain is all real numbers), but no real number squares to a negative — the output pool only holds balls with y ≥ 0. The set of all possible outputs is the **range**. In one line: the domain guards the entrance, the range guards the exit.

🎮 你来当扫描员（30 秒）🎮 Your turn: scan the line (30 seconds)

道理讲完了。拖滑块让扫描点扫遍整条数轴：绿色区间能进、红色区间被拦；扫完再答 4 道题。Theory done. Drag the slider so the scan dot sweeps the whole number line — green zones enter, red zones are stopped; then answer 4 questions.

一句话记住它：定义域 = 允许输入的全体；值域 = 能输出的全体。 Remember it in one line: the domain is all inputs allowed in; the range is all outputs that can come out.

找定义域两把尺：分母 ≠ 0、根号内 ≥ 0Two rulers for the domain: denominator ≠ 0, inside of a root ≥ 0 值域看机器能产出什么：x² 的 y ≥ 0，1/x 的 y ≠ 0The range is what the machine can produce: y ≥ 0 for x², y ≠ 0 for 1/x 先定入口（定义域），再看出口（值域），顺序不能反Find the entrance (domain) first, then the exit (range)

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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.