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Chapter 13: Problem 11

Find the domain of the function. \(g(x, y, z)=\frac{x}{y}-\frac{y}{z}+\frac{z}{x}\)

Short Answer

Expert verified
The domain is all \((x, y, z)\) where \(x \neq 0, y \neq 0, z \neq 0\).

Step by step solution

01

Identify the function components

The function given is \(g(x, y, z) = \frac{x}{y} - \frac{y}{z} + \frac{z}{x}\). It is composed of three rational expressions: \(\frac{x}{y}\), \(\frac{y}{z}\), and \(\frac{z}{x}\). The function is only defined when all denominators are non-zero.
02

Determine conditions for each denominator

For \(g(x, y, z)\) to be defined, we need to ensure that each of the denominators in the rational expressions is not zero:- The denominator of \(\frac{x}{y}\) is \(y\), so \(y eq 0\).- The denominator of \(\frac{y}{z}\) is \(z\), so \(z eq 0\).- The denominator of \(\frac{z}{x}\) is \(x\), so \(x eq 0\).
03

Combine conditions to find the domain

The domain of the function is where all conditions are satisfied simultaneously. Thus, the domain of \(g(x, y, z)\) is all possible triples \((x, y, z)\) such that \(x eq 0\), \(y eq 0\), and \(z eq 0\). In other words, \((x, y, z) \in \mathbb{R}^3\) except where any of \(x\), \(y\), or \(z\) is zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Expressions
In mathematics, rational expressions are fractions that involve polynomials. Just like numerical fractions, the value of a rational expression is undefined when its denominator is zero.
In simpler terms, any expression formed as a fraction with a numerator and a denominator is a rational expression, provided both are polynomials. The key point is ensuring the denominator does not equal zero to avoid undefined expressions.
In the exercise, you see this with expressions like \( \frac{x}{y} \), \( \frac{y}{z} \), and \( \frac{z}{x} \). The denominators \( y \), \( z \), and \( x \) must not be zero, or the expressions blow up into infinity, making the function undefined.
Denominators
Denominators play a critical role in determining where rational expressions are valid. Since a denominator being zero causes a fraction to become undefined, special care must be taken.
When dealing with denominators in mathematical functions, always ask:
  • What values of the variables make the denominator zero?
  • What conditions are necessary to keep the denominators non-zero?

In our exercise, the denominators \(y\), \(z\), and \(x\) should all not be zero. This ensures each of the fractions \( \frac{x}{y}, \frac{y}{z} \), and \( \frac{z}{x} \) keep the function \( g(x, y, z) \) defined and workable throughout its domain.
Multivariable Functions
Multivariable functions involve more than one variable and are represented in higher-dimensional spaces. This makes their analysis more intricate compared to single-variable functions, especially concerning their domain.
When determining the domain of such functions, it is crucial to consider the conditions that govern all variables simultaneously. In this case, \( g(x, y, z) = \frac{x}{y} - \frac{y}{z} + \frac{z}{x} \) involves three variables concurrently, expanding the complexity.
To assess the domain:
  • Check each variable involved in rational expressions, ensuring their denominators do not yield zero.
  • Identify potential points where any variable's denominator within the function might zero-out.
  • Combine these conditions: here, ensuring \(x eq 0\), \(y eq 0\), and \(z eq 0\) across the entire 3D space \(\mathbb{R}^3\).
This process eliminates locations where the function would otherwise be undefined, providing a precise, valid domain.

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Most popular questions from this chapter

The object distance \(p\), image distance \(q\), and focal length fof a simple lens satisfy the equation $$ \frac{1}{p}+\frac{1}{q}=\frac{1}{f} $$ Determine the minimum distance \(p+q\) between the object and the image for a given focal length.

Find the extreme values of \(f\) in the region described by the given inequalities. In each case assume that the extreme values exist. $$ f(x, y)=16-x^{2}-4 y^{2} ; x^{4}+2 y^{4} \leq 1 $$

Show that the surfaces \(z=\sqrt{x^{2}+y^{2}}\) and \(10 z=\) \(25+x^{2}+y^{2}\) have the same tangent plane at \((3,4,5)\).

Show that every line normal to the sphere \(x^{2}+y^{2}+z^{2}=\) 1 passes through the origin.

If \(f(x, y)\) is the amount of a commodity produced from \(x\) units of capital and \(y\) units of labor, then \(f\) is called a production function. If $$ f(x, y)=x^{\alpha} y^{\beta} \quad \text { for } x>0 \text { and } y>0 $$ where \(\alpha\) and \(\beta\) are positive constants less than 1, then \(f\) is called a Cobb-Douglas production function. a. Show that \(f(t x, t y)=t^{\alpha+\beta} f(x, y)\). b. If \(z=f(x, y)\), show that $$ \frac{1}{z} \frac{\partial z}{\partial x}=\frac{\alpha}{x} \quad \text { and } \quad \frac{1}{z} \frac{\partial z}{\partial y}=\frac{\beta}{y} $$ and that $$ x \frac{\partial z}{\partial x}+y \frac{\partial z}{\partial y}=(\alpha+\beta) z $$
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