Square Root Not A Function

Status
Not open for further replies.
mathdad

mathdad

Full Member
Joined
Apr 24, 2015
Messages
925
In his textbook, Sullivan makes the following statement concerning g(x) = sqrt{x}.

"g is not a polynomial function because g(x) = sqrt{x} = x^(1/2), so the variable x is raised to the 1/2 power, which is not a nonnegative integer."

This statement is slightly vague to me. Can someone explain the above statement a different way?
 
It's not that the square root is not a function, but that it is not a polynomial function.

"Polynomial" is defined as a sum of terms, each of which is a product of a constant (which can be any number, including 1) and a whole-number power of a variable (or variables). That is, a term is [MATH]ax^n[/MATH], where a is a real number and n is a non-negative integer. Examples of valid polynomial terms are [MATH]0.3x^5[/MATH], [MATH]\pi x^{101}[/MATH], and [MATH]x^{1000}[/MATH].

Since 1/2 is not an integer, [MATH]\sqrt{x} = x^{1/2}[/MATH] is not a polynomial. Neither is [MATH]\frac{1}{x} = x^{-1}[/MATH].

If you go further and ask, why we require the exponent to be a non-negative integer, that's because anything else (negative numbers or non-integers) would result in a function with different behavior, particularly in that the domain would no longer be all real numbers.
 
It is important to pay very close attention to definitions in math. Frequently, the basis for mathematical advance has been the formulation of an exact vocabulary. But the habit of defining things carefully and of then being consistent in their use is valuable everywhere.
 
JeffM said:
It is important to pay very close attention to definitions in math. Frequently, the basis for mathematical advance has been the formulation of an exact vocabulary. But the habit of defining things carefully and of then being consistent in their use is valuable everywhere.
Click to expand...

Why do you think I am asking for a clear definition here? I know that definitions in math are highly important.
 
Dr.Peterson said:
It's not that the square root is not a function, but that it is not a polynomial function.

"Polynomial" is defined as a sum of terms, each of which is a product of a constant (which can be any number, including 1) and a whole-number power of a variable (or variables). That is, a term is [MATH]ax^n[/MATH], where a is a real number and n is a non-negative integer. Examples of valid polynomial terms are [MATH]0.3x^5[/MATH], [MATH]\pi x^{101}[/MATH], and [MATH]x^{1000}[/MATH].

Since 1/2 is not an integer, [MATH]\sqrt{x} = x^{1/2}[/MATH] is not a polynomial. Neither is [MATH]\frac{1}{x} = x^{-1}[/MATH].

If you go further and ask, why we require the exponent to be a non-negative integer, that's because anything else (negative numbers or non-integers) would result in a function with different behavior, particularly in that the domain would no longer be all real numbers.
Click to expand...

Based on what you said, the following are not polynomial functions, right?

4x^(-5), 7t^(6/7), 5g^(pi), 7m^(9.8), 7u^(e)
 
JeffM said:
You wanted to know why something was said. It was said because of a definition, which you either ignored or did not understand.
Click to expand...

I did not ignore. I did not understand what Sullivan stated but it is all clear now.
 
mathdad said:
I did not ignore. I did not understand what Sullivan stated but it is all clear now.
Click to expand...
You did not understand that sqrt(x) was not a polynomial BECAUSE you either ignored the definition or did not understand it.

Again I'm telling you that JeffM is 100% asking that you know the definitions.

If someone asked me if an expression was or was not a polynomial I would ask the person what they thought a polynomial was.

There is no way that you really know the definition of a polynomial but don't know why something, like sqrt(x), is not a polynomial.
 
Jomo said:
You did not understand that sqrt(x) was not a polynomial BECAUSE you either ignored the definition or did not understand it.

Again I'm telling you that JeffM is 100% asking that you know the definitions.

If someone asked me if an expression was or was not a polynomial I would ask the person what they thought a polynomial was.

There is no way that you really know the definition of a polynomial but don't know why something, like sqrt(x), is not a polynomial.
Click to expand...

I understand what JeffM is trying to do. I appreciate his effort to help me but I want to make a few things clear to everyone:

1. I work overnight hours. Do you know what overnight does to the human body?

2. I only have ONE COMPLETE DAY to myself, which is Tuesday. How many hours in a day?

3. I am not a formal college classroom student. My college days ended in 1993.

4. I do not have a computer or laptop. Do you know how long it takes to type definitions and word problems using my tiny cell phone screen keyboard?

5. I am simply revisiting math learned long ago. I am not seeing this stuff for the first time.

6. I am not preparing for a state test.

7. I am having fun with my Sullivan textbook.

8. I mean no disrespect to JeffM or anyone else here.

9. I joined this site to find solution steps to math questions that I can solve if EASY directions are displayed.

10. When I say MOVING ON, I mean the following:

A. I figured out how to solve the problem.
B. I am no longer interested in terms of the problem.
C. I am tired of going back and forth getting no where.
 
mathdad said:
Why do you think I am asking for a clear definition here? I know that definitions in math are highly important.
Click to expand...
Your book already contains a clear definition of 'polynomial'. Pay close attention to the conditions that the definition places on the exponents. Cheers

?
 
Otis said:
Your book already contains a clear definition of 'polynomial'. Pay close attention to the conditions that the definition places on the exponents. Cheers

?
Click to expand...

Clear definition to you and other math people. Textbooks have their own unique language and jargon. I am moving on.
 
Mathdad,
You come here to get help from us which I think is a good thing to do. I do not think that you get to decide when to move on. I think that if someone here does not think that you understand things as well as they want you to understand it then you, out of respect to the volunteer helpers, should listen and try to see what they are saying.
I personally have asked questions here on the forum and received multiple replies. When I understood one reply but not the other I still tried my best to understand the other replies because I respect the knowledge that these helpers are trying to get me to see. And when I just can't understand the post, then I ask for further explanation.

In addition to learning how to see things in different ways I really think that you should answer all questions asked of you just out of respect. Who knows, maybe you will even learn something!
 
mathdad said:
Clear definition to you and other math people ...
Click to expand...
When you had read the definition of 'polynomial' and realized that something wasn't clear, you ought to have resolved your confusion before "moving on".

... I am moving on.
Click to expand...
That's not the hallmark of a serious person.

\(\;\)
 
Status
Not open for further replies.
Top