https://r2.erweima.ai/mathbot/image/1726078418388mxt4703d.jpg

The given function is:

\[
f(x, y) = \frac{\sqrt{(x-4)^2 + y^2} - 1}{xy}
\]

We are asked to determine the maximum domain of \(f(x, y)\) and create its graphical representation.

### Step 1: Determine the Domain

The domain of the function depends on several factors:
1. **Square Root:** The expression inside the square root, \((x-4)^2 + y^2\), must be non-negative:
   \[
   (x-4)^2 + y^2 \geq 0
   \]
   This is always true, as squares of real numbers are always non-negative.

2. **Division by \(xy\):** We must ensure that the denominator \(xy\) is non-zero, so we impose the condition:
   \[
   x \neq 0 \quad \text{and} \quad y \neq 0
   \]
   This means the function is undefined when either \(x = 0\) or \(y = 0\).

Therefore, the domain of the function is all pairs \((x, y)\) such that:
\[
x \neq 0 \quad \text{and} \quad y \neq 0
\]

In interval notation, the domain is:
\[
\mathbb{R}^2 \setminus \{x = 0 \text{ or } y = 0\}
\]

### Step 2: Graphical Representation

The graphical representation of this function involves analyzing the behavior of \(f(x, y)\) over its domain. Specifically, we would explore:
- Behavior near the axes where the function is undefined.
- Symmetry of the function (since it involves squares of \(x-4\) and \(y\), it might have some symmetry).
  
Would you like me to plot the graph of the function for better visualization?

---

### Expanding Questions:
1. What happens to the value of \( f(x, y) \) as \( x \) or \( y \) approach zero?
2. Can we find any asymptotes or critical points for this function?
3. How does shifting \( x \) by 4 affect the symmetry of the graph?
4. What is the significance of the term \(-1\) in the numerator?
5. How does the function behave for very large or very small values of \( x \) and \( y \)?

### Tip:
Always check for undefined points in a function, especially when dealing with fractions and square roots.

Considere la función f(x, y) = (sqrt((x-4)^2 + y^2) - 1) / (xy). Indique el dominio máximo de f y realice la representación gráfica.

Analyze the multivariable function f(x, y) = (sqrt((x-4)^2 + y^2) - 1) / (xy), including its domain and graphical behavior. This problem involves square roots, division by variables, and domain restrictions. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation