Функции, композиция, преобразования и обратные функции ⟦number⟧
Введено| English | Значение |
|---|---|
| composition/ˌkɒmpəˈzɪʃn/ | Combining functions by applying one function to the output of another. |
| one-to-one/wʌn tuː wʌn/ | взаимооднозначное соответствие |
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:
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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:
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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 ____.
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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.
Какой ответ подходит к данному случаю?
Вычислять значения и преобразовывать между формулами функций, таблицами и графиками
An unrestricted quadratic has a single-valued inverse on all real outputs.
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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.
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Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
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The order follows the stated process; check each stage before the next.