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수학동아 - 폴리매스
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함께 풀고 싶은 문제
창의력을 기를 수 있는 수학 문제 또는 퍼즐을 내는 곳입니다.
[창의 퍼즐] 함수의 성질
디듀우 2019.09.22 03:16 조회 646

실수 전체를 정의역과 치역으로 갖는 함수 f(x)와 모든 실수 c에 대해 다음이 성립하는 f(x)의 필요충분조건을 구하시오.

(1) f(x+c)=f(x)+c

(2) f(cx)=cf(x)

(3) f(x^c)={f(x)}^c

 

이 문제 어떠셨나요?

글쎄요

0

어려워요

0

  •  
    cube120 Lv.5 2019.09.22 09:56

    (1)조건에 c=x 을 대입하면

    f(2x)=f(x)+x

    (2)조건에 c=2 을 대입하면

    f(2x)=2f(x)

    따라서 두 식을 연립하면

    f(x)=x

    이 식은 조건 (3)도 만족한다.

    주어진 조건을 만족하는 함수는는f(x)=x 이다.

     

    댓글 작성하기 좋아요0 댓글수2
    •  
      디듀우 Lv.7 2019.09.22 18:22

      4개 조건을 모두 만족하는 함수를 구하는 것이 아니라, 4개 조건 각각에 대해 필요충분조건인 함수의 성질을 구하는 것입니다. 또한 c는 임의의 실수이므로 x라고 할 수 없습니다.

      좋아요0
    •  
      cube120 Lv.5 2019.09.23 09:16

      아 모든 실수 c라고 적혀있어서 유한확정값이 아닌줄 알았네요.

      좋아요0
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