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Math Help - Composite Function Domains

  1. #1
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    Composite Function Domains

    Hey, I'm having trouble trying to figure out this general word problem.

    7. Answer with brief Explanation or Counterexample. Is the domain of f \circ g contained in the domain of f?
    I want to say that the domain of the composite should be contained within the intersection of the range of f and the domain of g, but is this correct? And is this an acceptable way of showing that answer?

    D_{f \circ g}=E_{f} \cap D_{g} where E_{f}=range\ of\ f and D_{g}=domain\ of\ g

    What do you guys think?
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  2. #2
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    Quote Originally Posted by superduper View Post
    Hey, I'm having trouble trying to figure out this general word problem.

    I want to say that the domain of the composite should be contained within the intersection of the range of f and the domain of g, but is this correct? And is this an acceptable way of showing that answer?

    D_{f \circ g}=E_{f} \cap D_{g} where E_{f}=range\ of\ f and D_{g}=domain\ of\ g

    What do you guys think?
    Almost. What you give is a subset of the range of f and I believe you want the domain. The domain of f\circ g is the subset of the domain of f that is mapped into that intersection. And since that is necessarily a subset of the range of f, you don't need to say that. Oh, and you have the order of the functions reversed. The domain of g\circ f is f^{-1}(dom(g)). The domain of f\circ g is g^{-1}(dom(f)).

    The answer to the question given, is "No, the f\circ g is not necessarily contained in the domain of f, it is contained in the domain of g". A counterexample to the original statement is g(x)= -x, f(x)= \sqrt{x}. g\circ f(x)= \sqrt{-x} so its domain is the set of all non-[b]positive[/tex] numbers while the domain of f is the set of all non-negative numbers.
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  3. #3
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    Perfect, that makes a lot more sense. Thanks man
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