Domains, inverses and composition · Higher · 域、逆函数与复合 · 高阶
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| domain/dəˈmeɪn/ | 定义域 | dìng yì yù |
Which inputs are allowed?
- A square-root model returns a real output only for some inputs. Its formula alone does not specify a complete function.
- This lesson studies domain 定义域: The set of allowed inputs to a function.
Choose the mathematical structure
- State the domain and range. For an inverse, first ensure the function is one-to-one on its domain. Composition fg means apply g first, then f; the intermediate output must be an allowed input to f.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines domain? · 下列哪个描述正确定义了定义域？
The set of allowed inputs to a function. · 函数的允许输入集合。
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For f(x)=√(x-2), x≥2 and the range is y≥0. From y=√(x-2), x=y²+2. Thus f inverse(x)=x²+2 with x≥0. For g(x)=x+3, fg(1)=f(4)=√2.
Domains, inverses and composition · 定义域、反函数与复合函数
State the domain and range · 写出定义域和值域
Compare the model with the worked case and explain one change. · 对比模型与已解案例，并说明其中一处变化。
For f(x)=2x+3, find f inverse(11). · 已知 f(x)=2x+3，求 f⁻¹(11)。
Solve 2x+3=11, giving x=4. · 解方程 2x+3=11，得 x=4。
Test a tempting shortcut
- Squaring can introduce extraneous solutions. Restricting a parabola's domain is essential before claiming an inverse. A horizontal translation inside f has the opposite sign to the graph's movement.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every quadratic function on all real numbers has an inverse function. This claim is false. Explain which definition or assumption it violates.
For f(x)=2x+3 and g(x)=x², find fg(4). · 已知 f(x)=2x+3 且 g(x)=x²，求 fg(4)。
Apply g first: g(4)=16; then f(16)=2×16+3=35. · 先应用 g：g(4)=16；再应用 f：f(16)=2×16+3=35。
Every quadratic function on all real numbers has an inverse function. · 所有定义在全体实数上的二次函数都有反函数。
Squaring can introduce extraneous solutions. Restricting a parabola's domain is essential before claiming an inverse. A horizontal translation inside f has the opposite sign to the graph's movement. · 平方运算可能引入增根。在声称存在反函数之前，必须对抛物线的定义域加以限制。若 f 内部进行水平平移，其符号与图像的移动方向相反。
Interpret a new situation
- Check f(f inverse(x))=x on the inverse domain. Use a sketch to test whether a horizontal line meets the original graph more than once.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For f(x)=√(x-2), find f(11). · 已知 f(x)=√(x-2)，求 f(11)。
Substitute into the square root: √(11-2)=√9=3. · 代入根号内计算：√(11-2)=√9=3。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
Use this in your course
- 8300 · Higher · 3.2. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The set of allowed inputs to a function. Choose the relationship, show the method, check its assumptions and interpret the result.