Domain and Range of Multivariable Functions

Concept Explanation

1. Algebraic Restriction Rules in Several Variables

Just as in single-variable calculus, domain restrictions arise from algebraic operations that are undefined in the real numbers:

1. Rational Functions ($\frac{P}{Q}$): The denominator cannot equal zero:

   $$Q(x, y) \ne 0 \quad \text{or} \quad Q(x, y, z) \ne 0$$
   • In 2D, setting $Q(x, y) = 0$ identifies excluded boundary curves or lines (e.g., $y \ne \pm x$).
   • In 3D, setting $Q(x, y, z) = 0$ identifies excluded boundary surfaces (e.g., spheres $x^2+y^2+z^2 \ne 1$ or planes).

2. Even Radicals ($\sqrt[2n]{u}$): The radicand must be non-negative:

   $$u(x, y) \ge 0 \quad \text{or} \quad u(x, y, z) \ge 0$$
   • In 2D, this defines solid regions such as disks ($x^2 + y^2 \le r^2$) or half-planes ($y \le mx + b$).
   • In 3D, this defines solid volumes such as balls ($x^2 + y^2 + z^2 \le R^2$) or half-spaces.
   • If the even radical sits in the denominator ($\frac{1}{\sqrt{u}}$), the inequality becomes strictly positive: $u > 0$.

3. Logarithmic Arguments ($\ln(u)$): The argument must be strictly positive:

   $$u(x, y) > 0 \quad \text{or} \quad u(x, y, z) > 0$$
   • Boundary curves where $u = 0$ are strictly excluded (open regions / open half-spaces).

4. Trigonometric Functions ($\tan(u)$): Undefined where cosine is zero:

   $$u \ne \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z}$$

2. Topological Classification of Domain Regions in $\mathbb{R}^2$

When graphing multivariable domains, engineers classify regions using standard topological properties:

• Boundary Point: A point $(a, b)$ is a boundary point of $D$ if every disk centered at $(a, b)$ contains at least one point in $D$ and at least one point outside $D$.
• Interior Point: A point $(a, b)$ is an interior point of $D$ if there exists a disk centered at $(a, b)$ lying entirely inside $D$.
• Open Region: A region containing none of its boundary points (defined by strict inequalities like $<$ or $>$). Represented with a dashed boundary.
• Closed Region: A region containing all of its boundary points (defined by non-strict inequalities like $\le$ or $\ge$). Represented with a solid boundary.
• Bounded Region: A region that can be completely enclosed within a finite disk of radius $R$ centered at the origin.
• Unbounded Region: A region that extends infinitely in at least one direction (e.g., half-planes, quadrants).

Closed & Bounded (Solid Disk):        Open & Bounded (Dashed Disk):      Unbounded (Half-Plane):
         ***                                  ...                                 /
       *  D  *                              .  D  .                              /  D (all above)
         ***                                  ...                               /
     x^2 + y^2 <= R^2                     x^2 + y^2 < R^2                      / y >= mx + b
   (Solid boundary included)            (Dashed boundary excluded)           (Extends to infinity)

3. Systematic Range Determination

To find the range:

1. Identify the bounds on the internal expressions (e.g., $x^2 \ge 0$, $0 \le x^2 + y^2 \le R^2$).
2. Trace the transformation of these bounds through the outer function (square roots, fractions, exponentials).
3. Check extreme points (origin, domain boundaries) and asymptotic limits.

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Engineering Applications

• Structural Mechanics: Defining the safe operational geometry (domain) of an aerospace component where stress $\sigma(x, y, z)$ remains real and below yielding limits.
• Geotechnical Slope Stability: Bounding regional terrain coordinates $(x, y)$ where soil pore pressure remains within stable non-liquefaction thresholds.
• Fluid Piping Systems: Bounding volumetric flow rates and temperatures where fluid velocity equations do not encounter vacuum singularities or cavitation zones.

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Example Problems

Example 1: Domain and Range of a Radical Function of Two Variables

Find the domain and range of $f(x, y) = \sqrt{16 - x^2 - y^2}$. Classify the domain as open/closed and bounded/unbounded.

Solution

1. Domain Condition:
   $$16 - x^2 - y^2 \ge 0 \implies x^2 + y^2 \le 16$$
   • Set-Builder Notation: $D = \{(x, y) \in \mathbb{R}^2 \mid x^2 + y^2 \le 16\}$.
   • Geometric Description: A closed circular disk centered at $(0, 0)$ with radius $4$.
   • Classification: Closed (contains boundary circle $x^2 + y^2 = 16$) and bounded (contained inside radius $4$).
2. Range Analysis:
   • Since $x^2 + y^2 \ge 0$, the minimum of $x^2 + y^2$ is $0$ at the origin $(0, 0)$, giving maximum output $\sqrt{16 - 0} = 4$.
   • The maximum of $x^2 + y^2$ in $D$ is $16$ along the boundary, giving minimum output $\sqrt{16 - 16} = 0$.
   • Range: $R = [0, 4]$.

Example 2: Domain and Range of a Rational Function with Excluded Surfaces

Find the domain and range of $g(x, y, z) = \frac{1}{x^2 + y^2 + z^2 - 4}$.

Solution

1. Domain Condition:
   $$x^2 + y^2 + z^2 - 4 \ne 0 \implies x^2 + y^2 + z^2 \ne 4$$
   • Domain: $D = \{(x, y, z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 \ne 4\}$.
   • Geometric Description: All of three-dimensional space $\mathbb{R}^3$ except the spherical shell of radius $2$ centered at the origin.
2. Range Analysis:
   • Let $u = x^2 + y^2 + z^2 \in [0, \infty)$. The denominator is $d = u - 4 \in [-4, \infty) \setminus \{0\}$.
   • If $0 \le u < 4$, $d \in [-4, 0) \implies \frac{1}{d} \in (-\infty, -1/4]$. (At the origin, $g(0,0,0) = -1/4$).
   • If $u > 4$, $d \in (0, \infty) \implies \frac{1}{d} \in (0, \infty)$.
   • Range: $R = (-\infty, -1/4] \cup (0, \infty)$.

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Connections & References

• Parent Concepts: Multivariable Functions, Domain of Algebraic Functions
• Sub-Concepts: Level Curves and Contour Maps, Traces and Cross-Sections of Surfaces, Level Surfaces
• Course Sessions: Session 08 - Multivariable Functions