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I have a rather elementary question about multivariable calculus.

In a function $f(x,y)$; are the two values for the variables arbitrary chosen from a given set of numbers or are there any relationships or restrictions beyond that?

What I mean as an example: $f(x,y)=x^2+y$, where $x$ and $y$ are elements of all real numbers. Is there any further restriction to this or any combination rule for $x$ and $y$ that is just given in such a function ?

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    $\begingroup$ "or any combination rule for y and y that is just given in such a function" Did you mean "for x and y"? Also, to answer the question, No - both variables are viewed independently, they can both vary along all real numbers. $\endgroup$
    – John Doe
    Apr 13, 2018 at 14:00

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No. $x$ and $y$ can be arbitrary real numbers, as long as the expression makes sense. Sometimes the domain may be smaller - just as in the one variable case. The domain for $f(x) = \sqrt{x}$ isn't the whole real line. The domain for $f(x,y) = \sqrt{x^2 + y}$ isn't the whole plane.

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For any given function $f(x)$, the set acceptable values of $x$ on the real line is called the domain $\mathrm D_f$ of the funcion $f$, i.e., $$\mathrm D_f\subset \mathbb R$$ By 'acceptable', I mean the values of $x$ for which the function $f$ is defined, consider the example $$f(x)= \sqrt x$$ Clearly, $f(x)$ is defined $\forall x>0$. Therefore, $D_f=\{x:x>0,x\in \mathbb R\}$.

Similarly, for the function $f(x,y)$, the permissible values of $(x,y)$ make the domain $D_f\in\mathbb R^2$.

The values of $x$ and $y$ are such that they satisfy mathematical combination, which is defined by the function $f$.

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