Mathematics: models, functions and higher maths · 数学：模型、函数和高等数学
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| composition/ˌkɒmpəˈzɪʃn/ | 函数复合 | hán shù fù hé |
| independent/ˌɪndɪˈpendənt/ | 独立的 | dú lì de |
A decision before an answer
- A fee table can describe a linear model, while a savings table can describe exponential growth. The pattern matters more than the story.
- Your goal: Apply number and quantity, algebra and functions.
答案前的判断
- 费用表可描述线性模型，而储蓄表可描述指数增长。模式比故事背景更重要。
- 目标：应用数与量、代数和函数知识。
Read the relationship
- Preparing for higher mathematics covers number, algebra, functions, geometry and statistics/probability. Essential skills combine rates, units and proportions.
- Use coordinate and Euclidean geometry and trigonometry.
阅读关系
- 高等数学预备课程涵盖数与量、代数、函数、几何及统计/概率。核心技能包括比率、单位和比例的应用。
- 使用坐标几何、欧几里得几何和三角学。
If f(x)=3x−2, f(4) equals:
Substitute 4: 3×4−2=10.
Use the defining rule
- For a function, track the input and output. Composition applies the inner function first.
- Interpret statistics, probability and essential-skill models.
运用定义规则
- 对于函数，需追踪输入和输出。复合函数优先应用内部函数。
- 解读统计、概率及核心技能模型。
Right triangle legs 5 and 12: hypotenuse?
c²=25+144=169, hence c=13.
Check the conditions
- Coordinate geometry links slopes, distances and equations. Right-triangle ratios require the correct reference angle.
- Interpret statistics, probability and essential-skill models.
If f(x)=2x+1 and g(x)=x², then f(g(3))=f(9)=19. But g(f(3))=g(7)=49. Order matters. For a probability check, two independent fair coin tosses give P(two heads)=1/2×1/2=1/4.
检查条件
- 坐标几何连接斜率、距离和方程。直角三角形比值需使用正确的参考角。
- 解读统计、概率及核心技能模型。
若 f(x)=2x+1 且 g(x)=x²，则 f(g(3))=f(9)=19。但 g(f(3))=g(7)=49。顺序至关重要。对于概率验证，两次独立抛掷公平硬币得到 P(两次正面)=1/2×1/2=1/4。
The mean of 4,6,8 is ____.
Sum 18 divided by 3 values gives 6.
Apply the task format
- Model a context with stated assumptions, then check whether the result is sensible. A calculator supports computation but cannot choose a model for you.
- Interpret statistics, probability and essential-skill models.
Enhanced ACT maths uses four answer choices. A five-choice older item can teach a skill but should not define current test pacing.
应用题目格式
- 用明确假设对情境建模，然后检查结果是否合理。计算器支持计算，但不能替你选择模型。
- 解读统计、概率及核心技能模型。
增强版ACT数学使用四个选项。包含五个选项的旧题型可教授某种技能，但不应定义当前考试的节奏。
Which answer fits this case? · 哪个答案符合此案例？
Apply number and quantity, algebra and functions · 应用数与量、代数及函数
f(g(x)) always equals g(f(x)).
Function composition is generally not commutative.
Keep the distinctions
- composition 函数复合 — Applying one function to the output of another.
- independent 独立的 — One outcome does not change the other’s probability.
- Apply number and quantity, algebra and functions.
- Use coordinate and Euclidean geometry and trigonometry.
- Interpret statistics, probability and essential-skill models.
保持区别
- composition 函数复合 — 将一个函数应用于另一个函数的输出结果。
- independent 独立的 — 一个结果不会改变另一个结果的概率。
- 应用数与量、代数和函数知识。
- 使用坐标几何、欧几里得几何和三角学。
- 解读统计、概率及核心技能模型。
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.