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Homework Help: Ratio of functions, surjective (analysis course)

  1. Dec 21, 2009 #1
    1. The problem statement, all variables and given/known data
    let f: R->R be a continuous function
    Suppose k>=1 is an integer such that

    lim f(x)/x^k = lim f(x)/x^k = 0
    x->inf x->-inf

    set g(x)= x^k + f(x)

    g: R->R

    Prove that
    (i) if k is odd, then g is surjective
    (ii) if k is even, then there is a real number y such that the image of g is [y,inf)

    2. Relevant equations



    3. The attempt at a solution

    I am completely stuck at this all I can think of is x^k goes to infinity then the ratio of the functions can go to 0 if either f(x) goes to 0 or f(x) is a constant or f(x) goes to infinity slower than x^k (I am not sure about this)

    Any help will be very much appreciateve
     
  2. jcsd
  3. Dec 21, 2009 #2
    Because [tex]g[/tex] is continuous, you can use the intermediate value theorem. To show that [tex]g[/tex] is surjective, it is enough to show that [tex]g[/tex] becomes both very large and very large negative.

    To prove this part, you need to think about the qualitative behavior of [tex]f[/tex] and [tex]g[/tex]. The hypothesis on [tex]f[/tex] says that [tex]f(x)[/tex] is negligible compared to [tex]x^k[/tex] as [tex]|x|[/tex] becomes large. Therefore [tex]g(x)[/tex] should "behave almost like" [tex]x^k[/tex] as [tex]|x|[/tex] becomes large. Figure out a way to make this precise.
     
  4. Dec 21, 2009 #3
    Thank you for the reply.
    Where do I use the intermediate value theorem? Is it to prove surjective?
     
  5. Dec 21, 2009 #4
    Yes.
     
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