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What is the difference between these two expressions? Watch

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    In a tutorial, the tutor was talking about instances where a relation doesn't fulfil the conditions of a function. He mentioned y=\sqrt{x}, but this is a function, so he changed it to y=x^{\frac{1}{2}} and said this was a different expression to the first and wouldn't be a function.

    I asked my calculus lecturer and he gave a vague answer saying that my tutor had a point but, according to him, they represent the same thing. So what, if any, is the difference between the two expressions?
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    (Original post by Desmos)
    In a tutorial, the tutor was talking about instances where a relation doesn't fulfil the conditions of a function. He mentioned y=\sqrt{x}, but this is a function, so he changed it to y=x^{\frac{1}{2}} and said this was a different expression to the first and wouldn't be a function.

    I asked my calculus lecturer and he gave a vague answer saying that my tutor had a point but, according to him, they represent the same thing. So what, if any, is the difference between the two expressions?
    The only thing I can think of is that one is supposed to give both positive and negative answers (x^(1/2)), while the other is defined to be the positive root only (root x). This would fit in with the idea that the latter is a function (single valued), whereas the former is not (two possible answers for a given positive x).

    But I don't think this is absolutely clear unless this has been specified...
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    (Original post by Pangol)
    The only thing I can think of is that one is supposed to give both positive and negative answers (x^(1/2)), while the other is defined to be the positive root only (root x). This would fit in with the idea that the latter is a function (single valued), whereas the former is not (two possible answers for a given positive x).

    But I don't think this is absolutely clear unless this has been specified...
    Why would one return two values whilst the other returns one? Is it just because we define \sqrt to return only positive values?
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    (Original post by Desmos)
    Why would one return two values whilst the other returns one? Is it just because we define \sqrt to return only positive values?
    Yes, I think this is exactly it. That's why we need to write, for example, plus and minus root 2 as the solution of x^2 = 2. If root 2 could be positive or negative, the "plus or minus" would be superfluous.
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    (Original post by Desmos)
    In a tutorial, the tutor was talking about instances where a relation doesn't fulfil the conditions of a function. He mentioned y=\sqrt{x}, but this is a function, so he changed it to y=x^{\frac{1}{2}} and said this was a different expression to the first and wouldn't be a function.

    I asked my calculus lecturer and he gave a vague answer saying that my tutor had a point but, according to him, they represent the same thing. So what, if any, is the difference between the two expressions?
    Is the integral in your profile picture missing the dt and + c ?
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    Yeah, as another poster said, one gives a single value for each domain value, where as y=x^0.5 gives the positive and negative root and thus one is a mapping and one is a function.

    y= \pm \sqrt x = mapping
    as if you let x = 1 it will return 1 and -1
    y=\sqrt x = function
    if you let x=1 then it will return 1 only
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    As far as I know, x^{1/2} and \sqrt{x} are precisely the same thing and your tutor is wrong.
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    (Original post by Zacken)
    As far as I know, x^{1/2} and \sqrt{x} are precisely the same thing and your tutor is wrong.
    If you did it in a calculator yes, but that's just a limitation of a calculator
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    (Original post by M4cc4n4)
    If you did it in a calculator yes, but that's just a limitation of a calculator
    No, as set theoretic functions \mathbb{R}_{\geqslant 0} \to \mathbb{R}.
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    X^(1/2) refers to the principal square root of x. There is no difference.
 
 
 
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