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Function bijections help

  1. Mar 23, 2009 #1
    1. The problem statement, all variables and given/known data

    I really don't understand the question for this problem, could you please help me out? Thanks so much

    1.a,b are some real numbers. Give an example of a bijection from [0,1] to [a,b]
    2. Prove that all functions from [0,1] to [a,b] are bijections


    2. Relevant equations



    3. The attempt at a solution

    I think the example function could be something like Sqrt(1-x) + Sqrt(x). That makes its domain be [0,1]. But I don't know how to make it end up with the value of [a,b]
     
    Last edited: Mar 23, 2009
  2. jcsd
  3. Mar 23, 2009 #2

    Mark44

    Staff: Mentor

    Try a function like this--f(x) = c(x - h)--with domain restricted to [0, 1]. You'll need to figure out what the constants c and h need to be so that f(0) = a and f(1) = b.
     
  4. Mar 23, 2009 #3
    Hi, oh yes, your function works. All functions f(x) = mx + n are bijections. Thanks for your help.

    About the second one, do you think I should break it into 3 cases. With a,b < 0; a,b > 0; and a <0 ^ b>0?
     
  5. Mar 23, 2009 #4

    Mark44

    Staff: Mentor

    Before looking at your cases, you should first ask yourself whether it's true. "All functions" covers a lot of territory, including functions that are continuous as well as those that are discontinuous.
     
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