Section 4.1 - Functions of Several Variables

Section Objectives

1. Recognize a function of two variables and identify its domain and range.
2. Sketch or describe the graph of a function of two variables.
3. Recognize a function of three or more variables and identify its domain and range.
4. Identify and describe the level curves or surfaces of a function of several variables.

Functions of two variables

A function $f$ of the two variables $x$ and $y$ is a rule or correspondence that maps each ordered pair $(x,y)$ in some subset $D$ of $\mathbb{R}^2$ to a real number. The set, $D$, of all ordered pair inputs is called the domain of $f$. The set of all real number outputs is called the range of $f$.

This definition generalizes in obvious ways to functions of three or more variables. More on those later!

If the domain of a function is not explicitly given, we will assume the domain is the largest set of ordered pairs (or triples, etc.) that produce valid, real outputs. To determine these "implied domains," we must think about the types of operations applied by the function. With practice, domains are easy to determine, but finding ranges can be challenging.

Examples

• Find the domain and range of $f(x,y)=2x-9y+3$.

• Find the domain and range of $g(x,y)=\sqrt{1-x^2-y^2}$.

Graphs of two-variable functions

The graph of a two-variable function is a surface in 3-space. Specifically, the graph is the set of all points $(x,y,z)$ where $(x,y)$ is in the domain of $f$ and $z=f(x,y)$.

Just as you built up a library or tool kit of common graphs in 2-space, we'll do the same for surfaces. In fact, we are already familiar with planes, cylinders, and quadric surfaces.

Examples

• Sketch or describe the graph of $f(x,y)=2x-9y+3$.

​x
var("x,y")
plot3d(2*x-9*y+3, (x,-5,5), (y,-5,5))

• Sketch or describe the graph of $g(x,y)=\sqrt{1-x^2-y^2}$.

xxxxxxxxxx
var("x,y")
plot3d(sqrt(1-x^2-y^2), (x,-1,1), (y,-1,1), plot_points=(200,200))

• Sketch or describe the graph of $h(x,y)=x^2+y^2$.

x
var("x,y")
plot3d(x^2+y^2, (x,-2,2), (y,-2,2), mesh=True)

Level curves (a.k.a traces)

Suppose we are given a function $f(x,y)$. If $c$ is some number in the range of $f$, then the two-dimensional graph of the equation $f(x,y)=c$ is called a level curve of $f$. We've seen this idea before---when we were studying quadric surfaces, we called this level curve the trace associated with $z=c$.

Just as with the traces we studied before, identifying and sketching level curves can help us understand and visualize the graph of a two-variable function. A collection of various level curves is called a contour map.

Examples

• Sketch a contour map for the function $f(x,y)=x^2+y^2$.

xxxxxxxxxx
var("x,y")
contour_plot(x^2+y^2, (x,-2,2), (y,-2,2), contours=[0,1,2,3,4], fill=False, labels=True )

Functions of three or more variables

The ideas described above generalize to functions of three or more variables.

Examples

• Determine the domain of $\displaystyle f(x,y,z)= \frac{3x-4y+2z}{\sqrt{9-x^2-y^2-z^2}}$.

• Determine the domain of $\displaystyle g(x,y,t)=\frac{\sqrt{2t-4}}{x^2-y^2}$.

• For a function of three variables, $w=f(x,y,z)$, the graph is in 4-dimensional space. The surfaces defined by the three-variable equations $f(x,y,z)=c$, where $c$ is a number in the range of $f$, are called level surfaces. Describe the level surfaces of the function $g(x,y,z)=x^2+y^2+z^2$.