Functions of two variables

Definition: A function of the variables $x$ and $y$ is a prescription $z(x,y)$, which calculates for any combination of feasible values of the variables $x$ and $y$ a number, the function value.

All feasible values $D_1$ of $x$ and $D_2$ of $y$ together form the domain of the function.

The set of all possible function values is called the range of the function.

Remark: The function values $z(x,y)$ can be interpreted as the values of a variable. If we call this variable $z$, then $x$, $y$, and $z$ satisfy the equation
z & =z(x,y).
The variables $x$ and $y$ in $z(x,y)$ are called the independent or input variables} and the variable $z$ the dependent or output variable.

Example: A function of the variables $p$ and $q$ is for instance $C(p,q)=p^2 - pq + 10$.

Both $p$ and $q$ may take any value, which means that the domain of the function consists of all the possible combinations of all numbers. The independent variables are $p$ and $q$, the dependent variable is $C$.

It holds that $C(3,5)=3^2 - 3\cdot5 + 10 = 4$.