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Spivak chapter 3 question 6

  1. Nov 3, 2013 #1
    Hello, I have been backtracking recently on Spivak problems to ensure I am fluent and I'm doing this one and I just cannot solve it:

    For which numbers C is there a number X such that f(cx) = f(x). Hint: there are a lot more than you might think at first glance.

    I get this isn't really calculus but its in the book so I thought the responders here would be most useful.

    I appreciate all responses.
     
  2. jcsd
  3. Nov 3, 2013 #2
    I have moved forward while I wait for a response and have come to problem 9 with characteristic functions and it asks me to express A intersects B, X is either in A or B, and R - A ( all as subsets of C) in terms of [itex]C_A[/itex] and [itex]C_B[/itex].

    I don't know what it is asking me to do.
     
  4. Nov 3, 2013 #3
    What is [itex]f[/itex]? What's its domain? Is it continuous?

    If it's continuous and its domain includes zero, those might be useful properties. If there's also a point [itex]x>0[/itex] such that [itex]f(x)=f(0)[/itex], that might also be a useful property.
     
  5. Nov 3, 2013 #4
    For the second question, it's asking for algebraic ways of writing the statements given in terms of [itex]C_A,C_B[/itex]. For example, the statement [tex]A\neq\emptyset[/tex] is equivalent to the expression [itex]C_A\neq 0[/itex].
     
  6. Nov 3, 2013 #5
    Thanks. I didn't think of it that way.
     
  7. Nov 3, 2013 #6

    pasmith

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    Homework Helper

    Given c, all you need to do is choose x such that cx = x.
     
  8. Nov 4, 2013 #7
    The domain is x cannot equal -1. How does that help me? You could say that it is any number if x=0. Is that acceptable?

    For the second problem, I still don't see what you mean. How do I algebraically manipulate them?
     
  9. Nov 4, 2013 #8
    I have just realised this is in the wrong space but i don't know how to move it. Im sorry and feel free to move it.
     
  10. Nov 4, 2013 #9
    It turned out that C can be anything so i was right there. The second one is a bit tricky though. I will reread the text first.
     
  11. Nov 4, 2013 #10
    I see that [itex]C_A[/itex][itex]\bullet[/itex][itex]C_B[/itex] or [itex]C_A[/itex][itex]+[/itex][itex]C_B[/itex] could both be the answer for [itex]C_A\cap C_B[/itex] as both intersect with x. Is this right?How do i tell which it is?
     
  12. Nov 5, 2013 #11
    Economics nerd, you're saying that you can say A is not empty by writing [itex]C_A[/itex] is not 0 as, if its zero, x is not in A so A is empty.

    I see what you mean but could you please explain to me how these characteristic functions work. I get that C_A(x) means that there is an x for every A. However, I am having relating that to [itex]C_A \cup C_B[/itex]?
     
  13. Nov 6, 2013 #12
    Any response to my two questions?
     
  14. Nov 6, 2013 #13
    Also, in this context, what is the difference between adding, multiplying and subtracting in terms of the domain.
     
  15. Nov 6, 2013 #14
    Any thoughts?
     
  16. Nov 7, 2013 #15
    Is it normal to have issues with such problems early on, even if it is spivak?
     
  17. Nov 7, 2013 #16
    Anybody got any hints?
     
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