Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 27-32, describe geometrically the domain of each of the indicated functions of three variables. $$ f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}-16} $$

Step-by-Step Solution

Answer

The domain is the region outside or on the sphere of radius 4 centered at the origin.

Step 1: Understand the Function and Its Domain

The function given is \( f(x, y, z) = \sqrt{x^2 + y^2 + z^2 - 16} \), which is defined as long as the expression inside the square root is non-negative. Hence, to determine the domain of the function, solve the inequality \( x^2 + y^2 + z^2 - 16 \geq 0 \).

Step 2: Solve the Inequality

The inequality \( x^2 + y^2 + z^2 - 16 \geq 0 \) becomes \( x^2 + y^2 + z^2 \geq 16 \). This represents values of \( x, y, z \) for which the sum of their squares is greater than or equal to 16.

Step 3: Interpret Geometrically

The inequality \( x^2 + y^2 + z^2 \geq 16 \) describes the region outside or on the surface of a sphere centered at the origin with radius 4. The set of points \( (x, y, z) \) satisfying this inequality form the domain of the function.

Key Concepts

Function Domain

The domain of a function refers to the set of all possible input values (in this case, the coordinates \((x, y, z)\)) for which the function is defined. For the given function \(f(x, y, z) = \sqrt{x^2 + y^2 + z^2 - 16}\), it is crucial to ensure that the expression inside the square root is non-negative. This means solving the inequality \(x^2 + y^2 + z^2 - 16 \geq 0\).

To rephrase, the domain consists of all points \((x, y, z)\) where \(x^2 + y^2 + z^2 \geq 16\).
These are the input values that allow the function to output real numbers without involving the square root of a negative number. When identifying domains, you practice ensuring the function works smoothly across all intended inputs.

Geometric Interpretation

When examining multivariable functions, it's beneficial to visualize how the domain looks in geometric terms. This approach aids in understanding constraints and behaviors of complex functions. For the inequality \(x^2 + y^2 + z^2 \geq 16\), which outlines the domain, you can interpret it as all points located on or outside a sphere.

• At the center of this conceptual sphere is the origin (0, 0, 0).
• The radius of the sphere is \(4\), derived from \(\sqrt{16}\).

Geometrically, any point \((x, y, z)\) that satisfies this condition can be thought of as lying either on the boundary of the sphere (i.e., the surface) or further away from the origin. This sphere helps visualize constraints on how the function behaves, serving as a useful tool to conceptualize potentially complex mathematics in a more intuitive manner.

Inequalities in Mathematics

Inequalities like \(x^2 + y^2 + z^2 \geq 16\) play a vital role in defining domains in mathematical functions, particularly when dealing with multivariable scenarios. Such inequalities describe sets of conditions that offer insight into the possible space of solutions or domain.

In the context of the function \(f(x, y, z) = \sqrt{x^2 + y^2 + z^2 - 16}\), this inequality ensures non-negative values inside the square root, keeping the function defined real by identifying and excluding values that would result in nonsensical (negative) scenarios.

By representing spatial constraints or limits, inequalities help refine our understanding of where functions can reach or yield specific values. They are crucial tools in mathematics for clarifying and solving problems that involve limits, thresholds, or regions.