Domain and Range 3 variables

[math921]

I'm trying to find the domain and range of problems in 3 dimensional space or with 3 variables. The problem is their is an exponetial in the problem to

f(x,y)=y^2e^(-x^2)
Range? domain?

 

[pka]

The function \(\displaystyle f(x,y) = y^2 e^{ - x^2 }\) is a function \(\displaystyle \Re ^2 \to \Re ^1\).
Therefore, it is not 3-dimensional.

Can you use any ordered pair in the function?
So what is its domain?

It is never negative. WHY?
Can it be zero? HOW?
Then what is its range?

 

[stapel]

Note: If one considers the graph of the function, with f(x, y) = z, then the points on the surface formed would be of the form (x, y, z). In such a case, this could be viewed as being three-dimensional, much as f(x) = y could be considered two-dimensional due to the nature of its graph.

Eliz.

 

[pka]

Eliz, I completely disagree with that!
If one has f(x,y)=z then x and y are variables, but z is a constant!
In f(x,y)=z maps R<SUP>2</SUP> to the English Alphabet; it is constant z.

 

[stapel]

If one has f(x,y)=z then x and y are variables, but z is a constant! In f(x,y)=z maps R<SUP>2</SUP> to the English Alphabet; it is constant z.

Um... no, I don't think so....

The "z" is a variable, just like the "x" and the "y". Besides the fact that constants are generally "a", "b", and "c" (that is, from the front of the alphabet), I think I defined "z" to be another expression for "f(x, y)", analogous to "y" being another expression for "f(x)".

And just as solutions to "f(x) = (whatever formula)" are (x, y) points on the two-dimensional plane, solutions to "f(x, y) = (some other formula)" are (x, y, z) points in the three-dimensional space.

Sorry for the confusion.

Eliz.

 

[pka]

“The "z" is a variable, just like the "x" and the "y". Besides the fact that constants are generally "a", "b", and "c"” That is total nonsense! Where did you learn your set theory?

If you are saying that in vector calculus we sometimes write that z=x<SUP>2</SUP>+y<SUP>3</SUP> is a surface, then that is correct.
But this is a matter of function notation. The above example is a case f(x,y,z)=0 where f(x,y,z)=x<SUP>2</SUP>+y<SUP>3</SUP>−z. Sometimes this called a level curve.

 

[stapel]

Since the question referred to the domain of a function, I'm not sure where you're getting that the poster is looking for the solutions (the level curve) corresponding to some value of the function...?

In any case, while I fully respect your right to your opinion, I plan, in the interests of clear communication with the student, to stick with the customary usage.

Thank you for your understanding.

Eliz.

 

[pka]

Oh dear!
This is the 21th century! I think that I am the one who fully understands the question. The domain is the set of all ordered pairs of real numbers. The range of the function is the set of non-negative real numbers.

You are the one who brought in the possibly that this is a some sort surface.
There is absolutely nothing in the original question to suggest that it is anything more than a question about functions. I gave an answer about functions.