Chapter 8, Problem 1CRQ

Fill in the blanks.

1. The domain of a function f of two variables is a subset of the _______ -plane. The rule of f associates with each___ _____ in the domain of f one and only one ______, denoted by z = _______.

To determine

To fill: “The domain of a function f of two variables is a subset of the _______-plane. The rule of f associate with each ______ _____ in the domain of f one and only one _______, denoted by $z=\_\_\_\_$”.

Answer to Problem 1CRQ

“The domain of a function f of two variables is a subset of the xy-plane. The rule of f associate with each ordered pair in the domain of f one and only one real number, denoted by $z=f\left(x,y\right)$”.

Explanation of Solution

Suppose that f is a real valued function and consist of two variables x and y.

These variables are called independent variables.

The domain of the function is a set of ordered pairs of real numbers $\left(x,y\right)$.

Each ordered pair in the domain will associate with only one real number.

That real number is denoted as z, which is dependent on the values of x and y. So, z is called dependent variable.

Here $z=f\left(x,y\right)$ means that the value of f at the point $\left(x,y\right)$ is z.

Thus, “the domain of a function f of two variables is a subset of the xy-plane. The rule of f associate with each ordered pair in the domain of f one and only one real number, denoted by $z=f\left(x,y\right)$”.