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Thread: Please help me to find the domain

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    Please help me to find the domain

    Could someone please explain how to find the domain and range of: 1/square root (x^2+4). Without graphing. Thank
    Last edited by happy10; 19 Feb 2018 at 5:45 PM.

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    Senior Member sida1049's Avatar
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    Re: Please help me to find the domain

    The domain is all reals; x can take any real value.

    How do you work this out? Well, you start by asking yourself "are there any points or regions at which the function is undefined?" Since the function is a rational function (i.e. a fraction), an obvious place to start is by asking yourself for what values of x will the denominator be zero (and cause a problem)?

    But a quick glance at the denominator should tell you that the denominator can never be zero for any value of x; specifically, the denominator is always 2 or higher.

    Now, for any other possible points at which the function becomes undefined, note that there is a square root. Since we're considering only real numbers, recall that we cannot find the square root of a negative number. So you should now ask yourself for what values of x is x^2 + 4 < 0 ? The answer is none. So the square root doesn't cause any problems in the function.

    So there are no values of x such that the function will become undefined, and therefore the domain of the function is all reals.

    Hope this helps.

    Bachelor of Science (Advanced Mathematics) III, USYD

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    Re: Please help me to find the domain

    Oh ok I get it thanks so much. I’m not sure how to find the range though.

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    Senior Member integral95's Avatar
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    Re: Please help me to find the domain

    The Range would be



    You should think about what the lowest value you could have in your denominator, in that case it would be

    That lowest value would give you the highest value in the function (the maximum)

    Now think about the largest possible value in the denominator, that would be infinity since x could always increase without bound.
    This means the lowest value of the function is 0. (Now it actually doesn't get to the value 0, this way the range doesn't include that value)

    The reason it can't be negative is that when you take a square root, you can only take a square root of a positive number, which only gives a positive value.
    “Smart people learn from their mistakes. But the real sharp ones learn from the mistakes of others.”
    ― Brandon Mull

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    Re: Please help me to find the domain

    Ok thanks I understand how that works now, but then how would I find the range of 1/ squareroot(x^2+5x+6)? because there’s a 5x too

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    Senior Member integral95's Avatar
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    Re: Please help me to find the domain

    Quote Originally Posted by happy10 View Post
    Ok thanks I understand how that works now, but then how would I find the range of 1/ squareroot(x^2+5x+6)? because there’s a 5x too


    Since

    “Smart people learn from their mistakes. But the real sharp ones learn from the mistakes of others.”
    ― Brandon Mull

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