Functions of One or Two Variables

Question

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.

Answer 1

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:

• What is a function?