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The given function is:

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

### 1. **Domain of the function**

We need to determine the largest set of points \((x, y)\) for which the function is defined. This function has two major restrictions:

1. **The denominator**: 
   \[
   xy \neq 0
   \]
   This implies that both \(x \neq 0\) and \(y \neq 0\).

2. **The square root**: 
   The expression inside the square root must be non-negative:
   \[
   (x - 4)^2 + y^2 - 1 \geq 0
   \]
   Simplifying this inequality:
   \[
   (x - 4)^2 + y^2 \geq 1
   \]
   This represents the region outside (or on the boundary of) a circle with center \((4, 0)\) and radius 1.

Thus, the domain of \(f(x, y)\) is the set of points \((x, y)\) such that:
- \(x \neq 0\), \(y \neq 0\),
- \((x - 4)^2 + y^2 \geq 1\).

### 2. **Graphical representation**

To graph the function:
1. **Plot the circle**: The inequality \((x - 4)^2 + y^2 \geq 1\) describes a circle with center \((4, 0)\) and radius 1. The function is only defined outside or on this circle.
   
2. **Restrictions for \(x\) and \(y\)**: Exclude the lines \(x = 0\) and \(y = 0\) from the domain, as the function is undefined at these points due to division by zero.

Would you like a more detailed graphical explanation, or should I proceed with plotting this function?

---

Here are 5 related questions to deepen your understanding:

1. How can we handle cases where \(x = 0\) or \(y = 0\) in this type of function?
2. How would the domain change if the function involved other inequalities within the square root?
3. What happens to the function as \((x, y)\) approaches the boundary of the circle \((x - 4)^2 + y^2 = 1\)?
4. How would the graph of the function change if we modified the center or radius of the circle?
5. Can you describe other types of singularities or undefined points for this function?

**Tip**: When analyzing complex functions, always start by examining the domain and identifying key restrictions like denominators or square roots.

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

Explore the domain and graphical representation of the multivariable function f(x, y) = √((x - 4)^2 + y^2 - 1) / (xy). Learn about the restrictions due to the square root and division, and understand how to graph the function for its valid domain. Suitable for undergraduate mathematics students.

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