Problem 23 - APEX Calculus

Question

In Exercises \(23-26,\) give the domain and range of the functions of three variables. $$ f(x, y, z)=\frac{x}{x+2 y-4 z} $$

Step-by-Step Solution

Answer

Domain: \((x, y, z)\) where \(x + 2y - 4z \neq 0\). Range: all real numbers.

Step 1: Identify the Function

The function given is \(f(x, y, z) = \frac{x}{x + 2y - 4z}\). We need to find the domain and range of this function.

Step 2: Determine the Domain

The domain of a function is all the values for which the function is defined. Here, the function \(f(x, y, z)\) is undefined when the denominator is zero. Therefore, we need to solve the equation \(x + 2y - 4z = 0\) to find where the denominator is zero. This equation describes a plane in three-dimensional space, so the domain of \(f\) is all \((x, y, z)\) except those that lie on this plane:\[\text{Domain: } \{ (x, y, z) \,|\, x + 2y - 4z eq 0 \} \]

Step 3: Determine the Range

The range of a function is all possible output values. Since the numerator is \(x\), and as long as the denominator is not zero, any real number can be achieved by adjusting \(x\). Therefore, the range of \(f\) is all real numbers:\[\text{Range: } \mathbb{R} \text{ (all real numbers)} \]

Key Concepts

Understanding Functions of Three Variables

When dealing with multivariable calculus, functions of three variables are common. These functions take on the form \( f(x, y, z) \), where each input of the variables \( x, y, \) and \( z \) determines the output of the function. You can imagine this situation as a three-dimensional landscape, where every point in this space corresponds to a particular value of the function.

• Each variable influences the outcome of the function, making their interplay crucial for determining values.
• The function \( f(x, y, z) = \frac{x}{x + 2y - 4z} \) demonstrates this, by combining three inputs to produce one output.
• Functions like this can represent a wide variety of real-world phenomena, from thermodynamics to economics, where multiple factors influence results.

For functions of this nature, a key aspect is understanding how these variables interact, which leads us into discussing the domain and range.

Domain and Range in Multivariable Functions

The domain and range are essential in understanding any function, including those with three variables.
The domain consists of all possible input values that the function can accept. For \( f(x, y, z) = \frac{x}{x + 2y - 4z} \), the domain includes all sets of \((x, y, z)\) that do not make the denominator zero.

• By setting \( x + 2y - 4z = 0 \), we find the function's restriction: this equation forms a plane in 3D space, and points on this plane are where the function is undefined.
• Thus, the domain is every point except those on this specific plane.

The range is all possible output values the function can produce.
Since \( f(x, y, z) \) is not bound by particular restrictions on its outputs due to the nature of zero-avoidance in the denominator, it can potentially achieve any real number value.
Thus, the range is all real numbers, \( \mathbb{R} \).

Understanding Undefined Expressions

Recognizing the conditions leading to undefined expressions is vital in calculus. For functions of the form like \( f(x, y, z) = \frac{x}{x + 2y - 4z} \), undefined expression issues arise when the denominator becomes zero.

• This results in an undefined expression, as you cannot divide by zero.
• Understanding this aspect helps in mapping out excluded regions in the domain, which are crucial for graphing and interpretation.
• Solving \( x + 2y - 4z = 0 \) gives us a plane in the 3D domain where the function "blows up" or ceases to produce a finite value.

In practical terms, encountering undefined regions guides us in safely navigating and interpreting multivariable functions, avoiding potential pitfalls, and ensuring correctness in analysis.