Why is it that the square root function ( sqrt(x) ) only returns positive numbers? If you take the square root of, say, 16, isn't it incorrect to say that the result is only 4 instead of + or - 4? If it makes any difference, the problem came during a review of limits. The question was along the lines of what is the limit of f(x)=sqrt(x) as x approaches 4. The book says +2, but I thought it did not exist because sqrt(4) = +2 and -2, so the limit would not exist due to there being two different values. For that matter, the sqrt() function wouldn't really be a function if it has 2 values for a given point. My teacher said that the sqrt() function only returns positive values, so why is this? It doesn't make sense to me.
When talking about the output of a square root function, it's the absolute value that you graph and find the limit of. Since squaring 4 will equal 16 whether it's +4 or -4, all you need to say is that SQR(16) = 4; that -4 also works is a given. SQR is a true function because for every (positive) input, there is one output (which is ABS[SQR(x)]). In a way, SQR(x) gives two values, but in this case both output sets are equivalent (because when plugged into the inverse function f(x)=x^2, they do the same thing).
At least, that's how I understand it.
At least, that's how I understand it.
Dude! lol if f(x) = sqrt (x) and if it returns a positive and a negative answer that means for every x there is two Y values and you konw that you cant have a Function that returns two values for every x lmao. So basically that is why ur answer is positive. It is accepted that f(x) in this case is >0 or 0 and should we want to show the lower half we use f(x) = - sqrt (x) .
That is why you americans lack basic understanding. Things like this is laughable. You try to find limits and there is fundamental confusion on your behalf on what a function is.
That is why you americans lack basic understanding. Things like this is laughable. You try to find limits and there is fundamental confusion on your behalf on what a function is.
QUOTE (lol+Aug 29 2006, 10:50 AM)
Dude! lol if f(x) = sqrt (x) and if it returns a positive and a negative answer that means for every x there is two Y values and you konw that you cant have a Function that returns two values for every x lmao. So basically that is why ur answer is positive. It is accepted that f(x) in this case is >0 or 0 and should we want to show the lower half we use f(x) = - sqrt (x) .
That is why you americans lack basic understanding. Things like this is laughable. You try to find limits and there is fundamental confusion on your behalf on what a function is.
I'm glad we could each get a laugh out of one another's posts. LOLOLOLOL!
That is why you americans lack basic understanding. Things like this is laughable. You try to find limits and there is fundamental confusion on your behalf on what a function is.
I'm glad we could each get a laugh out of one another's posts. LOLOLOLOL!
the bottom graph i believe is f(x)^2 = x
ie
f(x) = +/- sqrt(x)
ie
f(x) = +/- sqrt(x)
Well, technically what LOL said is basically correct, although he said if very rudely in my opinion. If he really did l.h.a.o., his head probably came off with it.
There are two solutions to the equation x^2 = a, for a>0, of course. However, it is part of the definition of a function that it must associate only ONE output value with each input value. There are mathematical objects similar to functions, but which do allow multiple values; they are called "relations" to distinguish them from functions. This means that the square root operation is intrinsically a relation, not a function.
In order to make it into a function, it is necessary to remove one of the output values to make it single-valued. Which one is removed is mathematically arbitrary, but since humans need to communicate with each other, we should all agree to keep the same value and discard the same value. Therefore, BY CONVENTION, we arbitrarily decide to keep the positive value and discard the negative one.
Nothing bad would have happened if we had made the opposite choice, just some formulas would have different signs in them, but mathematics would come out the same. This is the human version of "spontaneous symmetry breaking" in physics. The underlying relation is symmetrical, but we have to break the symmetry to convert it into a function. As long as we all agree to break the symmetry in the same way, communication will be clear.
Hope this helps!
--Stuart Anderson
There are two solutions to the equation x^2 = a, for a>0, of course. However, it is part of the definition of a function that it must associate only ONE output value with each input value. There are mathematical objects similar to functions, but which do allow multiple values; they are called "relations" to distinguish them from functions. This means that the square root operation is intrinsically a relation, not a function.
In order to make it into a function, it is necessary to remove one of the output values to make it single-valued. Which one is removed is mathematically arbitrary, but since humans need to communicate with each other, we should all agree to keep the same value and discard the same value. Therefore, BY CONVENTION, we arbitrarily decide to keep the positive value and discard the negative one.
Nothing bad would have happened if we had made the opposite choice, just some formulas would have different signs in them, but mathematics would come out the same. This is the human version of "spontaneous symmetry breaking" in physics. The underlying relation is symmetrical, but we have to break the symmetry to convert it into a function. As long as we all agree to break the symmetry in the same way, communication will be clear.
Hope this helps!
--Stuart Anderson
depending on the nature of the question behind the math, sometimes you would wish to include the negative answers as well. Math is designed to compliment science. Math is nothing more than a tool that has been tested and works for science. Things that dont work, well, we discard them and put them to work in a philosophy class. Philosophy is a good class to take, but science, engineering, and math is what pays and philosphy gets to teach philosophy.
Anyway, if you are working on a scientific question about algee growth, you would only consider the rate of expansion in terms of an equation that would render a positive answer because you are studying growth. If you would be studying the growth while taking into consideration the dying algee cells, you would have a similar equation with more factors involved in the equation, and thus your math is complimenting the science, not the science complimenting the math. If you were studying the rate of death of algee cells, you would have an exponential that was negative and a solid understanding of math would advise you to rewrite the math accordingly. Rather than saying f(X)=X^2. you would say the rate of decay is equal to -f(X), where f(X)=X^2 and this would only yield negative results. The understanding comes in that math is designed to compliment science and how you ask the questions, math must be asked differently as well.
dwaynefries
Anyway, if you are working on a scientific question about algee growth, you would only consider the rate of expansion in terms of an equation that would render a positive answer because you are studying growth. If you would be studying the growth while taking into consideration the dying algee cells, you would have a similar equation with more factors involved in the equation, and thus your math is complimenting the science, not the science complimenting the math. If you were studying the rate of death of algee cells, you would have an exponential that was negative and a solid understanding of math would advise you to rewrite the math accordingly. Rather than saying f(X)=X^2. you would say the rate of decay is equal to -f(X), where f(X)=X^2 and this would only yield negative results. The understanding comes in that math is designed to compliment science and how you ask the questions, math must be asked differently as well.
dwaynefries
Does anybody know the answer to this
f®= sqrt r+6-6
f®= sqrt r+6-6
If we are dealing with positive square roots, the answers must remain positive. the improbable no.'s, i.e. ''Complex Numbers,'' deal with the square roots of negative no's... Numbers that don't exist. Keep them separate.
NeoNo.1
NeoNo.1
QUOTE (NeoNo.1+Aug 24 2007, 12:24 PM)
If we are dealing with positive square roots, the answers must remain positive.
Depends on the 'branch cut' you take. You are at liberty to define the square root to return a negative value, provided you're consistent.
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A square root has to be an absolute value.
Depends on the 'branch cut' you take. You are at liberty to define the square root to return a negative value, provided you're consistent.
QUOTE (NeoNo.1+Aug 24 2007, 12:24 PM)
the improbable no.'s, i.e. ''Complex Numbers,'' deal with the square roots of negative no's... Numbers that don't exist. Keep them separate.
The 'Reals', despite their name, are no more 'real' than the imaginary numbers. It's not a case of '5' exists but 'i' doesn't. Does '1/sqrt(2)' exist? What about '-1'? They are logical constructs and as valid as one another.
True.
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A square root has to be an absolute value.
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