Chapter 13, Problem 1GYR

To determine

Provide notes on a real-valued function of two independent variables and three independent variables with example.

Explanation of Solution

Suppose D is a set of n-tuples of real numbers $\left(x_1,x_2,\dots ,x_n\right)$.

A real-valued function f on D is a rule that assigns a real number $w=f\left(x_1,x_2,\dots ,x_n\right)$ to each element in D. The set D is the function’s range. The symbol w is the dependent variable of f, and f is said to be a function of the n independent variables $x_1$ to $x_n$. We also call the $x_j\text{'}\text{s}$ the function’s input variables and call w the function’s output variable.

If f is a function of two independent variables, that is the independent variables x and y and the dependent variable z.

Example:

The dependent variable z as a function of two variables is $z=\sqrt{y-x^2}$.

If f is a function of three independent variables, that is the independent variables x, y, and z and the dependent variable w.

Example:

The dependent variable w as a function of three variables is $w=\sqrt{x^2+y^2+z^2}$.