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(g of f)(x)

  1. Jun 7, 2004 #1
    Questoin...I have allways done these by evaluating (F 0f G)(x) ...then finding the domain of G and then useing a graphing calculator to find the domain of the whole thing as a function. My question is how do you algebracially fine the domain of the function after you find the domain of G int the case of (F of G)(X). If you can show me using an example where f=square root of x-2 and g= 2-square root(x)
     
  2. jcsd
  3. Jun 7, 2004 #2
    You can't find the domain of a function. It is usually given, implicitly or otherwise. Let h(x) = f(g(x)). What is the domain of h? What is the range? It should be obvious from the expression.
     
  4. Jun 7, 2004 #3

    arildno

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    These types of exercises should usually be interpreted as "what is the maximal domain this function can have?"
    In the given exercise, we have:

    [tex]f(x)=\sqrt{x-2},g(x)=2-\sqrt{x}[/tex]

    Hence, for f(g(x)), we need g(x)>=2

    This means: [tex]2-\sqrt{x}\geq{2}\to\sqrt{x}<=0[/tex]

    Hence, only a single x-value is eligible as the domain of f(g(x)), namely x=0,
    so that is the (maximal) domain.
     
    Last edited: Jun 7, 2004
  5. Jun 7, 2004 #4
    But this depends on whether f is defined on the reals. If the range of f is a subset of the complex numbers, then the 'maximal domain' is just the domain. This brings me back to my previous post.
     
  6. Jun 7, 2004 #5

    arildno

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    Sure enough, but these exercises rarely show up again once students have reached the level of learning about complex numbers..
    So, I stand by my answer as most probably correct :wink:
     
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