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Determine if these are functions

  1. Nov 18, 2012 #1
    Hello everyone,
    I just want to make sure I'm doing these problems correctly. Here they are

    Are the following functions?

    1. F : Z→Z where F(x) = 4/7x + 1
    Answer: Not a function. F(1) is not an integer.

    2. G : R→R where G(x) = {2x + 2 if x ≥ 0, x - 3 if x ≤ 0
    Answer: Not a function, because x is not well-defined. For G(0) there are two outputs.

    3. h : R→R where h(x) = { x^3 if x > 3, 2x - 3 if x ≤ 3
    Answer: Function

    Any suggestions are welcome.

    Thanks.
     
  2. jcsd
  3. Nov 18, 2012 #2

    micromass

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    That's correct.
     
  4. Nov 18, 2012 #3
    Thanks for the help!
     
  5. Nov 18, 2012 #4

    HallsofIvy

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    Just a nitpick- in 1 F certainly IS a function, just not from Z to Z.
     
  6. Nov 18, 2012 #5
    Good point. Maybe I should specify. Would something like this work?
    For Z→Z, F(x) = 4/7x + 1 is not a function.
     
  7. Nov 18, 2012 #6

    haruspex

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    For once, I disagree with Halls. F:Z→Z was part of the definition you were given, so the definition as a whole is not a valid definition of a function.
     
  8. Nov 18, 2012 #7

    micromass

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    I agree with this. The domain and the codomain are an essential part of a function.
     
  9. Nov 18, 2012 #8

    Ray Vickson

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    In 1: be careful, use parentheses. It makes a difference whether you mean F(x) = (4/7)x + 1 or F(x) = 4/(7x) + 1. In the first case F(x) is an integer whenever x is an integer multiple of 7, but in the second case F(x) is never an integer for any nonzero integer value of x.

    RGV
     
  10. Nov 20, 2012 #9
    Ray, since the domain and codomain are defined as Z→ Z wouldn't either case, F(x) = (4/7)x + 1 or F(x) = 4/(7x) + 1, still have to produce an integer for every integer x?

    Also, the function should have been written as F : Z→ Z where F(x) = (4/7)x + 1.
     
    Last edited: Nov 20, 2012
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