Is the square root of x and function or relation?

Question

There's a multiple choice question that asks to choose the answer that best describes the square root of x.

a) function
b) relation

The teacher tells me it's b) but I don't really understand why sqrt(x) is a relation over a functin?

Answer 1

The idea of a function is that you can go

f(5) = ?

and there should be one answer.



The technical reason square root isn't a function is it has two values, ie positive and negative of whatever the number is. The principle value, ie the one indicated by √5 is just the positive value, whereas an actual actual square root could be -√5 too. Calculators deal with this by only giving the principle value when you square root a number. The square roots of 5 can really only be correctly indicated by ±√5 which is two distinct numbers.

Answer 2

Are you given a domain and codomain also? y = √x with domain of all real numbers and codomain all real numbers isn't a function. A function needs to be defined for all numbers in the domain for it to be a valid function definition.

Answer 3

For me, what I call a "relation" is that: https://en.wikipedia.org/wiki/Binary_relation

For instance the set of pairs (a,b) in (R^+)^2 such that a^2=b is a relation (R^+ is the set of non-negative real numbers). This could be called "square root" if you want, but it's certainly less natural than defining "the square root" as a function in R^+ -> R^+.



If you want to generalise this definition outside R^+, there are plenty of possibilities to define what a "square root" is.



The simplest is certainly the algebraic definition: calling a a square root of b if and only if a^2=b (basically the same as the definition above but using R -or any group- instead of just R^+). In this case it takes the form of a relation, and is definitely not a function. However, a is not "the" square root of b, it's just "a" square root of b, there is no unicity. For instance 1 and -1 are both square roots of 1, just like in C we have that both i and -i are square roots of -1.



If we want some unicity, to be able to say that (almost) each number has one and only one square, we can define a cut in the complex plane. The standard way is to put the cut in R^-, so that we have an analytic function in C\R^- that we call the square root. Then "the" square root of 1 is 1 and only 1, "the" square root of i is (1+i)/sqrt(2), and so on, but the square root of -1 is not defined (although the square root of -1+0.00001i is close to i and the square root of -1-0.00001i is close to -i).

You can also decide to rotate the cut a bit further instead, somewhere in the third quadrant of C, so that the square root of 1 is still defined to be 1 but now the square root of -1 is defined as well and is i (but the square root of for instance e^(i 1.2 pi) would not be defined any more).



To go further than this you can use the concept of multivalued function and Riemann sheets. A multivalued function has no cut, it can be perfectly continuous everywhere, but it has a non-trivial topology instead, so that it is not a C->C function any more in the usual definition of what a function is.

That's probably way too high above your current maths level to be able to understand that, but what you should remember is that there are plenty of definitions. Choosing which one you want to use is up to you, as long as you are aware of it and you have correctly defined your choice. Because of this freedom to make an arbitrary choice (which is a very common thing in mathematics), I don't think this MCQ makes any sense, except if it has been preceded by a clear definition of what your teacher *decides* to call "the" square root.

Answer 4

Every function is a relation, so if you're allowed to choose only one answer then the question itself has problems.



As for whether "the square root" describes a function. It depends on the definition of "the square root" that you've been given. If I see the phrase "the square root" instead of "a square root", then I'd conclude that "the" implied a single value and that "the square root" is indeed a function. Not everyone will come to the same conclusion.



It's really not much of an issue. If your teacher says it's a "relation" then (s)he is thinking of what I'd call "a square root" where either -2 or 2 could be a square root of 4. Make a mental note of that and move on.



Personally, I put "the square root" in the same category as inverse trigonometric operations, and use the same resolution: When there are two or more answers, pick a principal value. Often, if the difference is important, as mathematical writer will use more specific language like "the principal square root" or "the square root function".



The majority use of the radical symbol √ is that principal value function, where the result is never negative. Look up "the quadratic formula" in any reference book and you're almost see something like:



x = [-b ± √(b² - 4ac)] / (2a)



If the √ were intended to represent a two-valued relation, there'd be no need for that ± symbol!

Answer 5

I would disagree with your teacher, but would not be willing to spend too much time arguing the point. Many teachers are misguided, but then again, many of them simply observe different definitions.



Here is my own take, for what it is worth.



The expression √(x) is a function.



The equation f(x) = √(x) defines function f.



The set of all ordered pairs (x, y) satisfying the equation y = √(x) is a relation.