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Math Help - Combinations of functions

  1. #1
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    Combinations of functions

    The question:

    Given that,
    f(x) = x + \frac{1}{\sqrt{x}}
    g(x) = \sqrt{x} - 2\sqrt{x}

    Find 6f + 3g.

    My attempt:

    6(x + \frac{1}{\sqrt{x}}) + 3(\sqrt{x} - 2\sqrt{x})
    6x + \frac{6}{\sqrt{x}} + 3\sqrt{x} - 6\sqrt{x}

    Apparently the answer is:
    6x + 3\sqrt{x}

    I'm not sure how they were able to simplify the expression that way. I'm sure it's something obvious that I'm overlooking. Any assistance would be great.
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  2. #2
    Senior Member eumyang's Avatar
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    Can you double check the original problem? I'm curious as to why g(x) was written as
    g(x) = \sqrt{x} - 2\sqrt{x}
    when you could combine like terms and rewrite g(x) as
    g(x) = -\sqrt{x}.
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  3. #3
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    That's just how the textbook wrote it.
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  4. #4
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    There is maybe a mistake in the textbook since, as already mentioned, you can simplify the expression of g by \displaystyle g(x) = \sqrt{x} - 2\sqrt{x} = -\sqrt{x} and since the answer is not the one you get

    I suggest that \displaystyle g(x) = \sqrt{x} - \frac{2}{\sqrt{x}}

    In this case \displaystyle 6f + 3 g = 6\left(x + \frac{1}{\sqrt{x}}\right) + 3\left(\sqrt{x} - \frac{2}{\sqrt{x}}\right) = 6x + 3\sqrt{x}
    Last edited by running-gag; July 16th 2010 at 11:39 AM. Reason: Typo correction
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  5. #5
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    Thanks. I guess I should check if there's errata for this text.
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