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함께 풀고 싶은 문제
창의력을 기를 수 있는 수학 문제 또는 퍼즐을 내는 곳입니다.
[창의 퍼즐] 정의역이 다른 두 함수
Amath 2021.07.27 05:58 조회 444
문제 푼 사람 0명
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어떤 정의역이 모든 정수의 집합이고, 공역도 모든 정수의 집합인 함수 y = x를 생각해 보자. 이 함수를 그리면 위 그림처럼 될 것이다. 이 같은 그래프를 그리는 정의역이 모든 실수의 집합인 함수 f(x)가 있을까?

 

우선, 어떻게 정의역이 모든 실수인데 x가 정수일 때만 그래프에 나타날 수 있을까? 그럴 수 없다. 

문제 1. 그렇다면 3차원 그래프, y축은 공역의 실수축, z축은 공역의 허수축, x축은 정의역인 그래프를 생각해 보자. 여기서 x축과 y축으로 이루어진 평면, 즉 공역의 실수인 경우만 그려진 그래프를 생각할 수 있다. 복소수는 표현하지 않는다고 할 때, 정의역이 모든 정수의 집합인 함수 y = x의 그래프와 같은 그래프를 그리는 정의역이 모든 실수의 집합인 함수 f(x)를 구하여라. 

 

문제 2. 이젠 좀 더 확장해 보자. 정의역이 모든 정수의 집합인 일차함수 y = ax + b의 그래프와 같은 그래프를 그리는 정의역이 모든 실수의 집합인 함수 f(x)를 구하여라. (복소수인 경우는 제외)

 

문제 3. 정의역이 모든 정수의 집합인 이차함수 y = ax + b의 그래프와 같은 그래프를 그리는 정의역이 모든 실수의 집합인 함수 f(x)를 구하여라. (복소수인 경우는 제외)

 

문제 4. 정의역이 모든 정수의 집합인 함수 y = g(x)와 같은 그래프를 그리는 정의역이 모든 실수의 집합인 함수 f(x)를 구하여라. (g(x)를 이용하여 나타내면 된다. )(복소수인 경우는 제외)

 

 

단, 수학의 어떤 기호도 사용할 수 있습니다. (삼각함수, 역삼각함수, 무한급수(리만 가설에서 나온 것도 되고), 절댓값, 번분수, 오일러 함수 등등 다 돼요. ) (이상한 건 말고)

 

(*제 문제는 풀이 필수인데, 이건 뭐 딱히 풀이가 필요하지 않은 것 같네요. )

이 문제 어떠셨나요?

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  •  
    cttg Lv.4 2021.07.28 19:25

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    f(x) = \lfloor x \rfloor

    정의역이 모든 실수이고 공역이 모든 정수인 함수가 존재하지 않나요?

    댓글 작성하기 좋아요0 댓글수1
    •  
      Amath Lv.8 2021.08.01 01:57

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      그렇죠. 그런데 정의역이 모든 실수이므로 모든 x에 대응되는 점이 하나씩은 있게 돼요. 그리고 그 함수 y = [x]를 그래프로 그려 보세요. (그런데 왜요?)

      좋아요0

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