Logic Behind a Function

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Jason76

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The idea of a function was invented and is now widely used. However, what was the purpose of thinking this stuff up to begin with? Two poloynomials are seen, one a function and the other not. So why is there a dividing line?
 
Jason76 said:
The idea of a function was invented and is now widely used. However, what was the purpose of thinking this stuff up to begin with? Two poloynomials are seen, one a function and the other not. So why is there a dividing line?
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You are not using terms correctly. Please type here specific examples of what you allege so that they may be addressed.
 
Jason76 said:
The idea of a function was invented and is now widely used. However, what was the purpose of thinking this stuff up to begin with? Two poloynomials are seen, one a function and the other not. So why is there a dividing line?
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So that instead of saying:

there exists a relation between a set of inputs(x) and a set of permissible outputs (y) with the property that each input is related to exactly one output

we can say

'y' is a function of 'x' - much shorter!!

Whole idea of mathematics is logically rigorous "short-cut".
 
I have always thought that the reason for making that distinction was related to the scientific idea that when different people run the same experiment, they should get the same result. You can also argue that we tend to expect that kind of thing even in non-science events. If you go to a store and see a number of different items with the same price, that is not at all surprising. But if you saw exactly the same item (same brand, same size, etc.) with two different prices, you would think something was wrong! If you look at a grade sheet for students in a particular class, finding different students with the same grade would not be unusual. But finding the same student with more than one grade would surprise you.
 
Jason76 said:
The idea of a function was invented and is now widely used. However, what was the purpose of thinking this stuff up to begin with? Two poloynomials are seen, one a function and the other not. So why is there a dividing line?
Click to expand...
The essence of the modern idea of a function is that it defines an unambiguous relationship with all permitted values of its arguments.

Thus \(\displaystyle \sqrt{1 - x^2}\text{ is a real-valued function for all real x such that } -1 \le x \le 1.\)

Any of the permitted values of x gives an unambiguous result.

But \(\displaystyle \pm \sqrt{1 - x^2}\) is not a function for any value of x except 0 because it is ambiguous.

It should not be surprising that mathematics seeks to avoid ambiguity.

Functional notation also lets relationships be described generically and compactly. Thus, for example, economists talk about an excess demand function d = f(p), where d and p are vectors. The number of terms in such a function may be numbered in the billions and could not, in practical terms, be written down even if all the terms were known, which they are not. So it is a very efficient way of asserting that a particular very complex relationship is unambiguous.

See http://en.wikipedia.org/wiki/History_of_the_function_concept

EDIT: Halls has an interesting idea that is remotely connected to my comment about ambiguity. There are relationships in the actual world that are unambiguous. There are other relationships that people expect or want to be unambiguous. The mathematicians developed the concept of function to make lack of ambiguity rigorous.
 
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JeffM said:
\(\displaystyle x^2 + y^2 = 1.\) I think that is a polynomial that is a function of neither x nor y.
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However, x2 + y2 = 1 is not a polynomial. It is an equation.

z(x,y) = x2 + y2 - 1

is a bi-variate polynomial function. For each unique input (x & y), the function returns a unique output (z).
 
Subhotosh Khan said:
However, x2 + y2 = 1 is not a polynomial. It is an equation.

z(x,y) = x2 + y2 - 1

is a bi-variate polynomial function. For each unique input (x & y), the function returns a unique output (z).
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Well that disposes of me. :grin:
 
JeffM said:
Well that disposes of me.
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Not so fast. According some your ideas may be right. There is no universal agreement.
Read this entry carefully. As you can see there a distinction between the use as an adjective or as a noun.

On this page there are minor differences with the above.

There does seem to be agreement that \(\displaystyle x^2-2xy+y^2\) is a polynomial (noun form);

\(\displaystyle x^2-2x+y=0\) is a polynomial function (adjective form).

As far as I can tell nobody thinks \(\displaystyle \dfrac{x^2+x+1}{x+1}\) is a polynomial.
 
pka said:
Not so fast. According some your ideas may be right. There is no universal agreement.
Read this entry carefully. As you can see there a distinction between the use as an adjective or as a noun.

On this page there are minor differences with the above.

There does seem to be agreement that \(\displaystyle x^2-2xy+y^2\) is a polynomial (noun form);

\(\displaystyle x^2-2x+y=0\) is a polynomial function (adjective form).

As far as I can tell nobody thinks \(\displaystyle \dfrac{x^2+x+1}{x+1}\) is a polynomial.
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pka and Subhotosh

Thank you both. I appreciate the concern with careful definition, and I actually glanced at the wiki page on polynomials before making post #7. If my vocabulary needs correction, I shall be happy to learn. If my usage is out of kilter, I suspect the same error in usage is the source of the OP's confusion.

Am I in error to say that x^2 + y^2 - 1 is a polynomial expression?

Am I in error to say that x^2 + y^2 - 1 does not define an implicit function of either y or x?

Am I in error to say that z = (x^2 + y^2 - 1) = g(x, y) is a multivariate function defined by a polynomial expression?

Am I in error to say that y^3 - x^2 + 1 = 0 does define an implicit function of x, namely \(\displaystyle f(x) = \sqrt[3]{x^2 - 1}\)?

If the answer to all those questions is "no," then the answer to the OP is that sometimes polynomial expressions define functions and sometimes they do not. The difference always revolves around the definition of "function" rather than the definition of "polynomial." If the answer to any of those questions is "yes," then I am just as confused as the OP.

I must admit that I view this as a side issue. What seems most important to me is that the OP understand that a function is a concise representation of an unambiguous relationship.
 
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