Dom 定义域和值域计算器（常见函数类型）Domain and Range Calculator (Common Function Types)

使用说明与常见问题How-to & FAQ

「定义域和值域计算器（常见函数类型）」是什么？What is the Domain and Range Calculator (Common Function Types) tool?

选择函数类型并输入参数，求定义域、值域、截距和关键点：一次函数、二次函数（顶点）、有理函数 (ax + b)/(cx + d)（渐近线）、平方根 a√(bx + c) + d、对数 a·ln(bx + c) + d、指数 a·bˣ + c 和绝对值 a|x − h| + k，结果用区间表示。本地处理。Pick a function type and enter its parameters to get the domain, range, intercepts and key points: linear, quadratic (vertex), rational (ax + b)/(cx + d) (asymptotes), square root a√(bx + c) + d, logarithm a·ln(bx + c) + d, exponential a·bˣ + c and absolute value a|x − h| + k, in interval notation. Processed locally.

如何使用「定义域和值域计算器（常见函数类型）」？How do I use the Domain and Range Calculator (Common Function Types) tool?

在「定义域和值域计算器（常见函数类型）」中，填写「函数类型」、「a」、「b」、「c」、「d」等参数，结果会随输入实时计算并显示在下方。In the Domain and Range Calculator (Common Function Types): Fill in "Function type", "a", "b", "c", "d"; results are recalculated instantly as you type.

定义域和值域是什么？What are domain and range?

定义域是 x 可以取的所有值，值域是函数 y 能取到的所有值。分母不能为 0、偶次根号下不能为负、对数的真数必须为正，这些都会限制定义域。The domain is every x you may use and the range every y the function reaches. A zero denominator, a negative value under an even root and a non-positive log argument all restrict the domain.

二次函数的值域怎么求？How is the range of a quadratic found?

先求顶点 y 值 c − b²/(4a)。a > 0 时开口向上，值域为 [顶点 y, ∞)；a < 0 时开口向下，值域为 (−∞, 顶点 y]。Find the vertex value c − b²/(4a). If a > 0 the parabola opens up and the range is [vertex y, ∞); if a < 0 it opens down and the range is (−∞, vertex y].

区间符号怎么读？How do I read interval notation?

方括号 [ ] 表示包含端点，圆括号 ( ) 表示不包含；∞ 总用圆括号；∪ 表示两个区间的并集。Square brackets include the endpoint and round brackets exclude it; ∞ always takes a round bracket; ∪ joins two intervals.

能举个计算实例吗？Can you show a worked example?

例如在「定义域和值域计算器（常见函数类型）」中填 函数类型 = 二次 y = ax² + bx + c，a = 1，b = 0，c = 0，结果区显示：函数：y = x²；定义域：(−∞, ∞)；值域：[0, ∞)；顶点：(0, 0)；对称轴：x = 0；y 截距：0；x 截距：0For example, entering Function type = Quadratic, a = 1, b = 0, c = 0 in the Domain and Range Calculator (Common Function Types) gives: Function: y = x²; Domain: (−∞, ∞); Range: [0, ∞); Vertex: (0, 0); Axis of symmetry: x = 0; y-intercept: 0; x-intercepts: 0

哪些输入会被检查？Which inputs are validated?

「定义域和值域计算器（常见函数类型）」会检查这些输入——「a」：必须填写；「b」、「c」、「d」：可留空；绝对值超过 10¹⁵ 的数也会被拒绝。不符合时结果区显示具体提示，不会给出数字。The Domain and Range Calculator (Common Function Types) checks its inputs — "a": required; "b", "c", "d": optional; numbers above 10¹⁵ in absolute value are rejected too. When a value fails, the result area names the problem instead of showing a number.

有哪些选项可以调整？What options can I adjust?

「定义域和值域计算器（常见函数类型）」的设置：「函数类型」可选 一次 y = ax + b、二次 y = ax² + bx + c、有理 y = (ax + b)/(cx + d)、平方根 y = a√(bx + c) + d、对数 y = a·ln(bx + c) + d、指数 y = a·bˣ + c等（默认 二次 y = ax² + bx + c）；「a」默认 1；「b」默认 0；「c」默认 0；「d」默认 0。每次生成结果都按当前设置处理。Settings in the Domain and Range Calculator (Common Function Types) — "Function type": Linear y = ax + b, Quadratic, Rational, Square root, Logarithm, Exponential, etc. (default Quadratic); "a" default 1; "b" default 0; "c" default 0; "d" default 0. The result always uses the current settings.

计算精度如何？How precise are the results?

「定义域和值域计算器（常见函数类型）」使用 JavaScript 双精度浮点数（约 15–16 位有效数字）；极大或极小的数值可能存在精度误差。The Domain and Range Calculator (Common Function Types) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.

「定义域和值域计算器（常见函数类型）」免费吗？需要注册吗？Is the Domain and Range Calculator (Common Function Types) free? Do I need an account?

「定义域和值域计算器（常见函数类型）」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Domain and Range Calculator (Common Function Types) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.