Mathematics: models, functions and higher maths
| English | 中文 | Pinyin |
|---|---|---|
| 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.
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.
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.
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.
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.
Which answer fits this case?
Apply number and quantity, algebra and functions
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.
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.