Problem 11
Question
Find the domain of the function. \(g(x, y, z)=\frac{x}{y}-\frac{y}{z}+\frac{z}{x}\)
Step-by-Step Solution
Verified Answer
The domain is all \((x, y, z)\) where \(x \neq 0, y \neq 0, z \neq 0\).
1Step 1: 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.
2Step 2: 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 \(x eq 0\).
3Step 3: 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.
Key Concepts
Rational ExpressionsDenominatorsMultivariable Functions
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.
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:
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.
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:
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 \(\mathbb{R}^3\).
Other exercises in this chapter
Problem 11
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Evaluate the limit. $$ \lim _{(x, y) \rightarrow(0,0)} \frac{\sin \left(x^{2}+y^{2}\right)}{x^{2}+y^{2}} $$
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Find the extreme values of \(f\) in the region described by the given inequalities. In each case assume that the extreme values exist. $$ f(x, y)=x^{3}+x^{2}+\f
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Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(x, y)=\frac{1
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