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Discrete math textbook problem

  1. Jan 7, 2009 #1
    1. The problem statement, all variables and given/known data
    find the domain and image of f such that
    f(x) = {(x,y) [tex] \in R \times R \vert x = \sqrt{y+3} [/tex]
    and domain and image of g such that
    g = {[tex] (\alpha,\beta) \vert \alpha is a person, \beta is a person, \alpha is the father of \beta [/tex]


    2. Relevant equations
    the domain and image are what would be expected of domain and image. No unstandard definitions.



    3. The attempt at a solution
    I think the answers in the back of the book are wrong. I will state my answers, the ones in the book have the opposite for domain and range (although it is a copyright 2009 book). It is also possible that I am wrong.

    okay for f, the function can be rewritten by substituing [tex] \sqrt{y+3} [/tex] for x.Then ,
    f = {[tex] (\sqrt{y+3},y) \vert y \in R [/tex]. This means that the domain will consist of values that are bigger or equal to -3. So, [tex] dom f = \{x \vert x \in R , x \geq -3 \} [/tex]. And following that, the image of f is the set of values that are greater than or equal to zero. [tex] im f = \{y \vert y \in R, y \geq 0 \}[/tex].


    For the second one. The domain is all people who are fathers, and the image is all people.(assuming that any person has a father where father denotes only a biological relationship).

    Any suggestions?
     
  2. jcsd
  3. Jan 7, 2009 #2

    Mark44

    Staff: Mentor

    For the first problem, and assuming that by domain the text's author means the set of x values, x has to be >= 0. The range is {y >= -3}.
     
  4. Jan 7, 2009 #3
    Ah yes, because [tex] x = \sqrt{y+3} [/tex], it would be impossible to have an x that is negative. Although, in case y is as small as possible, x can be zero. Thus, x >= 0. Now on the case of y, flip the radical and get that y = x^2 - 3. x >= 0, thus y >= -3. I see, I guess I fell into the trap the author wanted me to fall into.

    Now for the second one. Was I wrong here too? I think I am right, but now I am confused?
    let P = the set of all people
    let R(a,b) be the relation that a is the father of b.

    g = [tex] \{ (\alpha,\beta) \vert \alpha,\beta \in P, R(\alpha,\beta) \} [/tex]

    The domain of the function g should be the set of all people who are fathers. The image should be the set of all people who have fathers, in other words, all people. Is this correct?
     
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