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I know that a function of two variables is written in the form $f(x,y)$ = ....., where $x$ and $y$ don't have to appear explicitly.

The function $z$ = $93x^5 + 2y - 7x$, is that a function of one or two variables ? I know the domain must be a subset of the $x-y$ plane (real axis), so I would say that it is a function of two variables.

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    $f(x,y)=93x^5 + 2y - 7x$ is certainly a function of the two variables $x$ and $y$. The more interesting question is whether you would describe $f(x,y)=2y$ as a function of two variables if the domain is the $x,y$ plane – Henry Sep 12 '22 at 09:36
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    @Henry That's an interesting point. I would argue that $f(x,y)=2y$ is still a function of two variables because you need the entire point $(x,y)$ to extract the $y$. Consider, $f(x)=1$. It is constant with respect to $x$ but it is still a function of $x$. – John Douma Sep 12 '22 at 09:45
  • @Henry Good question! I would say: Yes of course, with $x$ free to take any value in the specified domain! (i.e. a plane) – user Sep 12 '22 at 11:32

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Yes your answer is correct, $z$ is a (real-valued) function of two variables indeed:

  • the value for $z$ is determined by two variable $x$ and $y$, that is $z=z(x,y)$,
  • and moreover at any pair $(x,y)$ corresponds one and only one value for $z$.

Both conditions are crucial for the definition of a function.

As a third ingredient, we also need to specify its domain and codomain, as for example (without restriction for the domain):

$$z: (x,y)\in \mathbb R^2 \to 93x^5 + 2y - 7x \in\mathbb R$$

Note that in this case the codomain corresponds also to the range.

Refer also to the related:

user
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