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## Main Question or Discussion Point

Some little detail has been giving me a head-ache for 1 of my problems.. And here is why.

Does ((-1)

Does ((-1)

^{2})^{1/2}equal 1 or -1? You decide...- Thread starter Nikitin
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- #1

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Some little detail has been giving me a head-ache for 1 of my problems.. And here is why.

Does ((-1)^{2})^{1/2} equal 1 or -1? You decide...

Does ((-1)

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CompuChip

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Because of your bracketing, there is nothing to decide. It is just 1, end of story ;)

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(1)

The function "square root" is not uniquely defined and requires a convention. I did not decide, this is the usual convention, I find it natural however.

If you intend to continue in mathematics, it will be best that you keep this example in mind : it will most probably be one of the first examples in the Riemann surface chapter.

I have never been as fast as CompuChip !

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Seriously, a problem like this thing ruined my entire evening.. I couldn't figure out what was wrong when I was simplifying a ridiculously long expression during my math homework...

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and yeh, I'l continue studying math when I go engineering.. I don't think they will teach us (imaginary numbers and unwanted stuff like that) such stuff until the final year tho..

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Hurkyl

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No. But if you asked the questionIsn't the better answer +-1??

What are the square roots of 1

the answer would be+1 and -1

The square root function simply isn't an inverse to the squaring function.

Incorrect simplification is a pervasive problem. People make this mistake with the trigonometric functions and their "inverses" a lot too.Seriously, a problem like this thing ruined my entire evening.. I couldn't figure out what was wrong when I was simplifying a ridiculously long expression during my math homework...

One of the more notorious mistakes is something like:

Oh, I have the equation

Can you spot the error?x(3+x) = x(5-x)

so I'll divide both sides by x3+x = 5-x

and so the answer is x = 1Many (most?) real-life problems break up into cases, and you have to solve all of the different cases (or, at least, admit to solving only a particular case). Unfortunately, many people seem to have great difficulty accepting this fact, and habitually ignore all except for the one -- e.g. they often ignore things like the fact that each positive real number has a negative square root or that variable expressions could be zero.

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- #8

BobG

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You need to read the wording of the responses and be sure not to gloss over the word "function". Functions have a specific definition.

As Hurkyl mentioned, that can cause problems with such things as trigonometry problems (quadrant resolution issues) since you often need all the possible answers. (Your calculator or a spreadsheet is only going to give you the answer to the function, not all possible answers.)

The most fun example of the problem with functions (and imaginary numbers) is graphing something such as [tex]y= -2^x[/tex].

The answer is:

-2 when x=1

4 when x=2

-8 when x =3

When does the graph cross zero?

There's a reason the exponential functions, [tex]f(x) = n^x[/tex], have to have a positive base other than 1.

Hmm, for some reason, my edits to latex seem to exist in some other dimension than visible posts. They show up in preview, but not in the post.

As Hurkyl mentioned, that can cause problems with such things as trigonometry problems (quadrant resolution issues) since you often need all the possible answers. (Your calculator or a spreadsheet is only going to give you the answer to the function, not all possible answers.)

The most fun example of the problem with functions (and imaginary numbers) is graphing something such as [tex]y= -2^x[/tex].

The answer is:

-2 when x=1

4 when x=2

-8 when x =3

When does the graph cross zero?

There's a reason the exponential functions, [tex]f(x) = n^x[/tex], have to have a positive base other than 1.

Hmm, for some reason, my edits to latex seem to exist in some other dimension than visible posts. They show up in preview, but not in the post.

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- #9

Hurkyl

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yeh, obviously x=0 is an answer as well in your question. I am no newb (not a master tho, obviously :p). I know that stuff can have multiple answers, but this problem I brought is kind of different.

Bob, hurkly; I am still not understanding, why is the correct way from bottom-to-top and not from top-to-bottom? This is just some simple curiosity.. It just seems that the only answer is, though,: "it just is so, so accept that!"

And trust me hurkly, that thing I was simplifying (the 1st derivative of a function of the type f(x)=u^(0.5) + p^(0.5).. and it didn't even look simple in its base-form) was not a "simple expression", it was a beast from hell.

This entire (1^2)^(1/2) problem was too much for mathcad, so I couldn't check if I had done any mistakes or not. That caused me great problems and finally brought me to this forum :)

Bob, hurkly; I am still not understanding, why is the correct way from bottom-to-top and not from top-to-bottom? This is just some simple curiosity.. It just seems that the only answer is, though,: "it just is so, so accept that!"

And trust me hurkly, that thing I was simplifying (the 1st derivative of a function of the type f(x)=u^(0.5) + p^(0.5).. and it didn't even look simple in its base-form) was not a "simple expression", it was a beast from hell.

This entire (1^2)^(1/2) problem was too much for mathcad, so I couldn't check if I had done any mistakes or not. That caused me great problems and finally brought me to this forum :)

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BobG

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So, some constraints have to be set since some equations can have two (or more) values that correspond to a given input.The mathematical concept of a function expresses the intuitive idea that one quantity (the argument of the function, also known as the input) completely determines another quantity (the value, or the output). A function assigns exactly one value to each input of a specified type.

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BobG

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No.

I assume you're asking if you could write two functions which happen to look identical to each other and say one equals -1 and the other 1?

Not unless you're redefining your constraints, in which case you're comparing apples in Washington to oranges in Finland. (And, yes, I know Washington is a state and Finland is a country - that's half the point.)

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If you mean, "Can a function have two different outputs for the same input?", then no. It's part of the definintion of a function that it has only one output for a given input. So [itex]f(x) = \sqrt{x}[/itex] defines a function [itex]f[/itex] which gives the positive square root of a real number [itex]x[/itex]. But this function isn't the inverse of the function [itex]g[/itex] defined by [itex]g(x) = x^2[/itex]. In fact, if [itex]g[/itex] is a function from the real numbers to the real numbers, it has no inverse function.

Something else to watch out for: sometimes the term "square root of x" refers to the positive square root, that is, [itex]\sqrt{x} \equiv x^{1/2}[/itex], while other times, the "square root of x" means "a real number, y, such that y

http://en.wikipedia.org/wiki/Square_root

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Let f map the set of nonnegative real numbers to itself, such that f(x) = xSome little detail has been giving me a head-ache for 1 of my problems.. And here is why.

Does ((-1)^{2})^{1/2}equal 1 or -1? You decide...

[tex]f \circ g \, (x) = (x^2)^{1/2} = \sqrt{x^2} = |x|.[/tex]

The value of this function on x is called the "absolute value of x". An equivalent way of defining the absolute value of x is to say:

If x < 0, |x| = -x,

otherwise |x| = x.

The absolute value of 1 is 1, since 1 is not less than 0.

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I meant, are there functions that have a different way of processing input? I am basically wondering if the "+1" answer to my question is ALWAYS right, i.e. none consider anything else correct.If you mean, "Can a function have two different outputs for the same input?", then no. It's part of the definintion of a function that it has only one output for a given input. So LaTeX Code: f(x) = \\sqrt{x} defines a function LaTeX Code: f which gives the positive square root of a real number LaTeX Code: x . But this function isn't the inverse of the function LaTeX Code: g defined by LaTeX Code: g(x) = x^2 . In fact, if LaTeX Code: g is a function from the real numbers to the real numbers, it has no inverse function.

Something else to watch out for: sometimes the term "square root of x" refers to the positive square root, that is, LaTeX Code: \\sqrt{x} \\equiv x^{1/2} , while other times, the "square root of x" means "a real number, y, such that y2 = x", where y can be positive or negative.

http://en.wikipedia.org/wiki/Square_root

Like 1+1=2.

Sorry for my English...

Rasal, heh don't worry.. I obviously know stuff like that! I always provide the +- y answer unless it would be illogical (like in a practical assignment).

Heh, I am starting to feel slightly stupid after asking this question.. :P

Edit: Yeah, thinking of it as an absolute-value makes sense though this is different from things like sqrt X = +-y (where +-y is the correct answer, only +y is wrong).. I mean, you can get either 1 or -1 depending on how you read it..

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- #17

Mark44

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For example, [itex]\sqrt{4}[/itex] = 2, not +/- 2.

It is true that 4 has two square roots, 2 and -2, but 2 is the

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No, there is no function which has more than one different way of processing a particular input. By definition, a function associates only one element each of its output set (codomain) with every element of its input set (domain). If a relation associates more than one output with a single input, then that relation is not a function.I meant, are there functions that have a different way of processing input?

Yes, ((-1)I am basically wondering if the "+1" answer to my question is ALWAYS right, i.e. none consider anything else correct.

Like 1+1=2.

Sorry for my English...

More stupid not to ask ;-)Rasal, heh don't worry.. I obviously know stuff like that! I always provide the +- y answer unless it would be illogical (like in a practical assignment).

Heh, I am starting to feel slightly stupid after asking this question.. :P

Edit: Yeah, thinking of it as an absolute-value makes sense though this is different from things like sqrt X = +-y (where +-y is the correct answer, only +y is wrong).. I mean, you can get either 1 or -1 depending on how you read it..

Compare the questions:

(1) If x

(2) What is 4

The answer to the first question is 2 or -2. The answer to the second question is 2.

g(x) = y such that y

f(x) = x

(As Mark44 mentioned, another name for the positive square root is "the principle square root".)

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- #19

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Just to take up this point, since others have done the main question to death.Also why is the other way around (eliminating the exponents first) wrong?

Perhaps you are thinking of differentiation rules for 'function of a function' which does indeed proceed from the outside to the inside.

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HallsofIvy

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I am no expert so this is just for discussion.

Does ((-1)^{2})^{1/2}equal 1 or -1? You decide...

Let us say we we have the equation x

x = + ((-1)

x = - ((-1)

Your question is in the form of solution 1 so the positive solution has already been chosen.

When we say 1

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Instead ofWhen we say 11/2 we are saying perform theof raising 1 by the power of (1/2) which is subtly different to asking what are the possible roots of the equation x2=1.function

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Which it can't, so question solved, thanks :)

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What a cryptic thing to say.Instead ofuse operation.function

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