Functions, composition, transformations and inverses
| English | 中文 | Pinyin |
|---|---|---|
| composition/ˌkɒmpəˈzɪʃn/ | 复合函数 | fù hé hán shù |
| one-to-one/wʌn tuː wʌn/ | 一一对应 | yī yī duì yìng |
A decision before an answer
- f(g(x)) applies g first. Reversing that order can change the value, formula and domain.
- Your goal: Evaluate and convert between function rules, tables and graphs.
Read the relationship
- A function gives one output for each allowed input. A graph passing a vertical line at two different output points is not a function of x. Domain lists allowed inputs; range lists attained outputs. Convert a table to a rule only when the relationship is supported, and interpret each variable’s role.
- Track input order and domain through composition.
f(x)=3x+1, g(x)=x-2. f(g(5)) equals:
g(5)=3, then f(3)=10.
Use the defining rule
- In f(g(x)), the inner function g acts first and its output must be in f’s domain. For f(u)=1/(u-1) and g(x)=x²+1, composition gives 1/x² and excludes x=0. Domain checking belongs to the original functions as well as the simplified result.
- Find an inverse after choosing a one-to-one domain.
The inverse of f(x)=4x-8 is:
Solve y=4x-8 for x=(y+8)/4.
Check the conditions
- An output change f(x)+k shifts vertically by k; f(x-h) shifts right by h. An outside multiplier changes output scale, while an inside multiplier changes how quickly the input reaches old points. Identify a known point to verify the transformation instead of guessing direction from a sign.
- Find an inverse after choosing a one-to-one domain.
f(x)=2x+3 and g(x)=x² give f(g(x))=2x²+3, but g(f(x))=(2x+3)². For h(x)=(x-2)² restricted to x≥2, the inverse is 2+√x with domain x≥0. A vertical shift of h by 4 has vertex (2,4); the unrestricted quadratic would not have a single-valued inverse.
The vertex of (x-5)²+2 has x-coordinate ____.
The horizontal shift is right by 5.
Apply the task format
- An inverse exchanges input and output and requires a one-to-one relationship on the chosen domain. Solve y=f(x) for x, then exchange names. A quadratic must usually be restricted to one side of its vertex. The inverse function f⁻¹ is not the reciprocal 1/f. Check composition in both directions on the allowed domain.
- Find an inverse after choosing a one-to-one domain.
Preserve composition order and the restricted inverse branch. Superscript -1 in function notation does not mean a reciprocal.
Which answer fits this case?
Evaluate and convert between function rules, tables and graphs
An unrestricted quadratic has a single-valued inverse on all real outputs.
Two different inputs can share an output; a branch restriction is needed.
Keep the distinctions
- composition 复合函数 — Applying an inner function before an outer function.
- one-to-one 一一对应 — A function in which distinct allowed inputs have distinct outputs.
- Evaluate and convert between function rules, tables and graphs.
- Track input order and domain through composition.
- Find an inverse after choosing a one-to-one domain.
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.