Problem 78 - Calculus Early Transcendentals

Question

Find the domains of the following functions. Specify the domain mathematically and then describe it in words or with a sketch. $$h(x, y, z)=\sqrt[4]{z^{2}-x z+y z-x y}.$$

Step-by-Step Solution

Answer

Based on the step-by-step solution, describe the domain of the function h(x, y, z) in words. The domain of the function h(x, y, z) can be described as follows: In a three-dimensional space, the function is defined when x equals y, for all values of z; when x is not equal to y and x is zero, the function is defined for z greater than or equal to 0 or z less than or equal to -y; and when x is not equal to y and y is zero, the function is defined for z greater than or equal to 0 or z less than or equal to -x. The domain is infinite as there is no maximum value for x, y, and z within these constraints.

Step 1: Identify the constraints of the function

The function is given by: $$h(x, y, z)=\sqrt[4]{z^{2}-x z+y z-x y}.$$ The only constraint for the function to be defined is that the expression inside the fourth-root must be non-negative, i.e., $$(z^2-xz+yz-xy) \geq 0.$$

Step 2: Rearrange the inequality

We have the inequality: $$(z^2-xz+yz-xy) \geq 0.$$ Rearranging the terms, we get: $$z^2 -(x-y)z - xy \geq 0.$$

Step 3: Analyze and find constraints

It's challenging to solve this inequality analytically. However, we can use some observations to derive constraints for x, y, and z that make the inequality true. 1. If \(x=y\), the inequality becomes \(z^2 \geq 0\), which is always true (as \(z^2\) will always be non-negative). So, when \(x=y\), the function is defined for all values of z. 2. If \(x\neq y\), let's consider the case when either \(x\) or \(y\) is equal to 0. Let's say \(x=0\), then the inequality becomes \(z^2 + yz \geq 0\), which is true if either \(z \geq 0\) or \(z \leq -y\). Similarly, when \(y=0\), we get the same constraints for x and z.

Step 4: Describe the domain

Based on our analysis, we can describe the domain of the function: 1. When \(x = y\), the function is defined for all values of \(z\). 2. When \(x \neq y\) and \(x = 0\), the function is defined for \(z \geq 0\) or \(z \leq -y\). 3. When \(x \neq y\) and \(y = 0\), the function is defined for \(z \geq 0\) or \(z \leq -x\). In words, the function is defined in a three-dimensional space where the restrictions on x, y, and z are those mentioned above. The domain is infinite since there is no maximum value on the coordinate axes that x, y, and z can take with these constraints.

Key Concepts

Inequality Constraints

When dealing with multivariable functions, inequality constraints are fundamental in determining where a function is defined. In our given function \( h(x, y, z) = \sqrt[4]{z^2 - xz + yz - xy} \), this constraint ensures the expression under the fourth root remains non-negative, i.e., \( z^2 - xz + yz - xy \geq 0 \). This inequality constraint prevents the root from becoming an undefined complex value.

To understand how to set these constraints, let's consider what each part of the inequality signifies:

• An unknown dependence: The variables \(x\), \(y\), and \(z\) create interdependencies, affecting each other's permissible range of values.
• Non-negativity requirement: The expression must be greater than or equal to zero, to keep the function real and defined across the domain.

By analyzing and rearranging the terms, we can better visualize and test where the expression remains non-negative and hence applicable for real-world problems.

Function Definition

The definition of a function in mathematical terms is crucial as it determines the set of allowable input values (domain) and output values (range). With our function \( h(x, y, z) = \sqrt[4]{z^2 - xz + yz - xy} \), the expression is embedded in a three-variable context.

The **function definition** helps us establish scenarios where the function is applicable:

• Variables and Relationships: Each variable \((x, y, z)\) plays a vital role. The function expression combines these to define specific conditions for its existence.
• Limiting Expressions: As with our fourth-root expression, it may have conditions such as positive values to maintain real number outputs.

Understanding function definitions allows students to identify crucial mathematical components and analyze their relevance across different problems, giving insight into how they define the operational space (domain).

Multi-dimensional Analysis

Analysis in multiple dimensions can seem daunting initially, but it's essential for comprehending complex functions involving several variables. Examining \( h(x, y, z) \) highlights how each variable contributes within a multi-dimensional space.

Key considerations in multi-dimensional analysis include:

• Spatial Understanding: Recognizing that \(x\), \(y\), and \(z\) represent separate axes in a 3D space is fundamental when visualizing their interactions.
• Interdependencies: How one parameter affects others is analyzed through constraints derived from their collective relationships, like \( z^2 - xz + yz - xy \geq 0 \).
• Domains as Regions: The analysis focuses on defining regions in space where the function is defined, which are influenced by inequality constraints. For instance, if \(x = y\), the potential values for \(z\) simplify the problem to be always defined.

Through multi-dimensional analysis, students can better interpret the function's behavior and visualize possible solutions within the allowable domains, enhancing problem-solving techniques in real-world contexts.