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Section 8.1 Graphs and Properties of Multivariable Functions
Objectives
Understand the concept of multivariable functions, especially of the form
\(z=f(x,y)\)
Understand the domain and range of a multivariable functions
Determine whether a representation/graph can be expressed with one coordinate as a function of the others
Subsection 8.1.1 Domain and Range
Figure 8.1. A screenshot lesson on plotting and domains of multivariable functions.
In our study of multivariable functions we have learned that \(z\) acts as the dependent variable and \(x,y\) are the independent variables. Intuitively, \(x,y\) are the inputs of the function and \(z\) is the output of the function. In this way, we have
\begin{equation*}
z=f(x,y).
\end{equation*}
As you may recall, functions have a domain. A simple way to think about this is the domain is the set of points \((x,y)\) that can be plugged into the function \(f\text{.}\) The rules you know and love from the previous classes still hold:
Do not divide by 0.
Do not take the square root (or fourth root, etc.) of a negative number.
Do not take the log of a nonnegative number. (A number that is 0 or negative.)
Of course, there are others, but this is a basic list. Since the domain of
\(f\) is a set of
\((x,y)\) points, the domain can be graphed in 2d on the
\(xy\) -plane. You may want to review
Section 3.4 before proceeding.
Activity 8.1 . Domain of Multivariable Functions.
Consider the function given by
\begin{equation*}
f(x,y) = \sqrt{36-9x^2-4y^2}
\end{equation*}
(a) Which of the following points would be in the domain of the function?
\(\displaystyle (-1,-1)\)
(b) Write an inequality that is satisfied for all
\((x,y)\) in the domain of
\(f\text{.}\)
(c) Graph the domain of the function
\(f\) by graphing the inequality from the previous part. Confirm your results on the tool below.
(d)
Repeat the previous parts for
\begin{equation*}
g(x,y) = \frac{1}{\sqrt{36-9x^2-4y^2} }\text{.}
\end{equation*}
(e) Compare the domain
\(f\) and the domain of
\(g\text{.}\) What is the same? What is the different?
Activity 8.2 . Domain and Range of Multivariable Functions.
Give the domain AND range for each of the following functions. Give your answers using set notation.
\(\displaystyle f(x,y)=\sqrt{x-y}\)
\(\displaystyle f(x,y)=\sqrt{1-y^2}\)
\(\displaystyle f(x,y)=ln(xy)\)
\(f(x,y)=x\) if \(x^2+y^2\leq 1\)
\(f(x,y)=ln(|xy|+1)\) for \(y\geq0\)
Exercises 8.1.2 Exercises
For the following, graph the domain of the function given. Use the Desmos tool above to confirm your results.
1.
\(f(x,y) = \sqrt{2y-x^2+2}\)
2.
\(f(x,y) = \frac{1}{\sqrt{y^2-4x^2+16}}\)
3.
\(f(x,y)= \frac{1}{2x-y+3}\)
4.
\(f(x,y) = \log(x^2+y^2-9) \)
Subsection 8.1.3 MVF: Tables and Graphs
Activity 8.3 . Writing one coordinate as function of the others.
(a) For each equation, state whether
\(z\) is a function of
\(x\) and
\(y\text{.}\) You should write a brief justification for your answer to each part.
\(\displaystyle z^2=x^2+y^2\)
\(\displaystyle z=\sqrt{x^2 y^2}\)
\(\displaystyle z^2=x^2 y^2\)
\(\displaystyle z=4\)
\(\displaystyle |z-x|=4\)
\(\displaystyle z=\begin{cases} x^2y \amp y \geq 0 \\
xy^2 \amp x\geq 0
\end{cases}\)
\(\displaystyle z=\begin{cases} xy^2 \amp y \geq x+2 \\
x^2y \amp y\leq x+2
\end{cases}\)
(b) For each graph, state whether
\(z\) is a function of
\(x\) and
\(y\text{.}\) You should write a brief justification for your answer to each part.
