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Homework Help: Multivariable domain

  1. Dec 3, 2013 #1
    1. The problem statement, all variables and given/known data
    I'm having a problem with solving the domain of |x^2+y^2|<=|z^2|.

    2. Relevant equations

    3. The attempt at a solution
    From what I got this should be separated into two expressions:
    x^2+y^2<=z^2, and x^2+y^2>=-z^2. The later doesn't have a real solution because of the negative root. The first one should then be z>=+-sqrt(x^2+y^2). I can understand that z>=+sqrt(x^2+y^2) is above the positive cone(inside), but where is z>=-sqrt(x^2+y^2) supposed to be? I'm thinking above the negative cone(outside), but I'm not sure.
  2. jcsd
  3. Dec 3, 2013 #2


    Staff: Mentor

    x, y, and z are all real numbers, right?

    If so, you can get rid of the absolute value symbols, since x2 + y2 ≥ 0 for any real numbers x and y. Similarly, z2 ≥ 0 for any real number z.
  4. Dec 3, 2013 #3
    Maybe should have put the part that was troubling me in the question.
    So basically, where is z>=-sqrt(x^2+y^2) supposed to be?
  5. Dec 3, 2013 #4


    Staff: Mentor

    If you strip out the unnecessary stuff, your inequality is x2 + y2 ≤ z2.

    First off, look at the equation x2 + y2 = z2. This is part of your solution set. What does it look like?

    Once you have that, then tackle the rest, which is x2 + y2 < z2. The absolute smallest that x2 + y2 can be is 0.
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