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Math Help - Composite Functions Question

  1. #1
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    Composite Functions Question

    Hope I've posted this question in the right place- I think I did.. But anyway.

    Let a be a positive number, let f : [2,∞) → R, f (x) = a − x and let g: (−∞, 1] → R,
    g(x) = x^2 + a. Find all values of a for which f ◦ g and g ◦ f both exist.

    I've no clue how to do this could someone show me?
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  2. #2
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    This question is all about domain.
    g\circ f demands that f(t) is in the domain of g.
    Whereas, f\circ g demands that g(t) is in the domain of f.
    Can you see that a=3 works BUT a=0 does not?

    Now work on it.
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  3. #3
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    Yes.. but where do you get the figure 3 from? You don't just randomly try out numbers do you ?
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  4. #4
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    No, he is not suggesting that you "try numbers at random" but he is giving two examples so you can see what is happening. We are given that the domain of f is not "all real numbers" but "all real numbers greater than or equal to 2". In order that f(g(x)) exist, we must have g(x)\ge 2. That says x^2+ a\ge 2 where, remember, x must be less than or equal to 1. What must a be so that g(1)= 1+ a\ge 2?

    In order that g(f(x)) exist, f(x) must be less than or equal to 1. That is, f(x)= a- x\le 1. But x must be greater than or equal to 2. What must a be so that f(2)= a- 2\le 1?
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  5. #5
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    Oh now it makes sense. So a = 2 or 3.
    Thanks guys
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  6. #6
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    It also could be 2.5.
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