What Do Domain and Range Mean in Multivariable Functions?

[Substance D]

hey,

hope this is the right spot for this...

im curious to know what Domain and Range refer to in a multvariable function. I understand what it means in a single variable (x is usually domain, y is usually range), but when it gets to Multivariable, it doesn't make sense to me. Domain seems to be the same, but what does Range refer to?

Thanks

 

[quasar987]

Even in a single variable function, x is not the domain and y is not the range. A function, in the most general sense, is an ordered triplet of sets (A, B, f) that we denote by f:A-->B, where f is some subset of A x B such that \forall a \in A, \exists !b\in B \ \mbox{such that} \ (a,b)\in f. A is called the domain and B the codomain of the function. The range of the function is the subset of B denoted by f(A) and defined by f(A)=\{b\in B:\exists a\in A \ \mbox{with} \ (a,b)\in f\}.

This definition however is not very easy to understand. It is easier to think of f not as a set but as a "rule" that associates to every a in A an unique b in B. If (a,b) is in f, we write f(a)=b. A is the domain, B is the codomain and the range f(A) is the set of all elements b of B for which there actually is an a in A such that f(a)=b.

To give a concrete exemple in the case of a real function of several variables, consider f:\mathbb{R}^2\rightarrow \mathbb{R} where the "rule of association" is f(x,y)=sin(xy). Then R² is the domain, R is the codomain and the range is [0,1].

 

[Substance D]

thanks!

but, um... I am going to have to do some more reseach into this becuase I didnt really understand your answer

 

[quasar987]

If you can verbalize what it is you find confusing, perhaps I can help you further?

Have you checked wikipedia? It has some nice illustrations and plenty of explaining.

domain: http://en.wikipedia.org/wiki/Domain_(mathematics)#Domain_of_a_function

codomain: http://en.wikipedia.org/wiki/Codomain

range: http://en.wikipedia.org/wiki/Range_(mathematics)