関数、逆分支および合成
導入済み| English | 意味 |
|---|---|
| bijection/baɪˈdʒekʃn/ | 双射 |
| involution/ɪnvəˈluːʃn/ | 対合 |
A decision before an answer
- A reversible encoding must recover exactly one input. A formula alone cannot tell you whether recovery is possible: its allowed inputs and outputs matter.
- Your goal: Distinguish injectivity and surjectivity using the stated domain and codomain.
Read the relationship
- A function assigns exactly one output to each allowed input. Injective means equal outputs force equal inputs; surjective means every element of the stated codomain is reached. The map x↦x² from R to [0,∞) is surjective but not injective. From [0,∞) to [0,∞) it is both, hence bijective. From [0,∞) to R it remains injective but fails surjectivity. Keep domain, image and codomain separate when testing each claim.
- Choose and verify an inverse branch by composing in both directions.
For f:[2,∞)→[−5,∞), f(x)=(x−2)²−5, what is f⁻¹(y)?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
The input restriction x≥2 forces x−2≥0, so select the positive square-root branch.
Use the defining rule
- An inverse function reverses a bijection. Solve y=f(x) for x, then use the original domain to choose a branch. For f(x)=(x−2)²−5 on x≥2, the inverse is 2+√(y+5) on y≥−5. The minus branch would return inputs outside the chosen domain. Check f⁻¹(f(x))=x for allowed x and f(f⁻¹(y))=y for allowed y; one unchecked composition can conceal a domain error.
- Trace repeated function composition while preserving the domain.
For g(x)=1/(1−x) on R excluding 0 and 1, what is g⁸(3)?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
The orbit is 3, −1/2, 2/3, then 3 again. Eight is congruent to two modulo three.
Check the conditions
- The inverse graph reflects the original graph across y=x; it does not take reciprocals of output values. A self-inverse function, or involution, satisfies f(f(x))=x wherever the composition is defined. Both −x on R and 1/x on R excluding zero are involutions. A strictly increasing involution on an interval must be the identity: if f(x)>x, increasingness gives f(f(x))>f(x)>x, and the analogous argument rules out f(x)<x.
- Trace repeated function composition while preserving the domain.
For f(x)=(x−2)²−5 with x≥2, f(5)=4 and f⁻¹(4)=2+√9=5. The unrestricted quadratic would have two inputs, −1 and 5, giving output 4. For g(x)=1/(1−x), the orbit 3→−1/2→2/3→3 has period three, so g⁸(3)=2/3; the count concerns compositions, not eighth powers.
For h(x)=−x on R, the value of h(h(7)) is ____.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
Negation twice returns the original input; this map is an involution.
Apply the task format
- Iteration fⁿ means repeated composition, not the power (f(x))ⁿ. Calculate the first few compositions and check their domains before looking for a period. For f(x)=1/(1−x) on R excluding 0 and 1, f²(x)=(x−1)/x and f³(x)=x. The image stays in the same allowed domain. Therefore reduce an iteration count modulo three. A displayed formula equal to x after cancellation does not restore forbidden inputs.
- Trace repeated function composition while preserving the domain.
Changing a codomain changes surjectivity. Reflection across y=x is an inverse graph; reciprocating y-values is a different operation. Do not cancel away excluded inputs.
このケースに当てはまる答えはどれか？
指定された定義域と値域を用いて、単射性と全射性を区別する
The function x↦x² from [0,∞) to R is surjective.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
No negative element of the codomain is reached. The image is only [0,∞).
Keep the distinctions
- bijection 双射 — A function that is both injective and surjective between its specified sets.
- involution 对合 — A function whose composition with itself is the identity on its domain.
- Distinguish injectivity and surjectivity using the stated domain and codomain.
- Choose and verify an inverse branch by composing in both directions.
- Trace repeated function composition while preserving the domain.
Match each term with its precise meaning in this lesson.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
難易度 3/5 · 200 コイン + 今日初回完了時に小ボーナス付与
The order follows the stated process; check each stage before the next.