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Domain / range of this function

  1. Sep 22, 2011 #1
    1. The problem statement, all variables and given/known data

    I don't remember the exact question right now, but it was something like this:

    f(x) = sqrt(4 - x^2)

    I needed to give the domain / range of it, and also symmetry I believe.


    2. Relevant equations



    3. The attempt at a solution

    I know this is a half circle after being graphed, but how can I show the domain and range in a math way?

    For domain, I know 4 - x^ must >= 0, so I can solve for it that way.
    But what about range?
    Can I also just isolate x and then get sqrt with y inside and restrict to real numbers?
     
  2. jcsd
  3. Sep 23, 2011 #2
    Formally, you should say: if "y" is an element in the domains, then there's an x, so that:

    [itex]\sqrt{4 - x^{2}}= y[/itex]
    That already means that y [itex]\geq[/itex] 0, since the square root always gives non-negative values.
    Squaring the equation:
    =>
    x2 = 4 - y2

    This equation only has a solution if the right side is positive (or zero). therefore:
    4 - y2 [itex]\geq[/itex] 0
    Check out what inequality you get from that.

    Of course you need to combine it with y [itex]\geq[/itex] 0 (squaring equations usually leads to extra solutions) with the inequality you got.

    A little less formally, but probably valid - you could use the graph of the function. By finding the absolute maximum and minimum of the function in it's closed range [-2,2] and noting that the function gets any values between them, being continuous in that segment - you can find your range.
     
  4. Sep 23, 2011 #3
    [tex] \sqrt4 = \pm 2 [/tex]
     
  5. Sep 23, 2011 #4
    The square root function is defined to be the positive root of a number, unlike the operation of taking a root from a number.
     
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