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We are given the function:

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

We need to find the maximum domain for this function and create its graphical representation.

### Step 1: Analyze the Square Root Terms
Both square root expressions place restrictions on the domain of \( f(x, y) \).

- First term: \( \frac{\sqrt{1 - x^2 - \frac{y^2}{4}}}{x - y} \)

For the square root to be real, the argument inside must be non-negative:

\[
1 - x^2 - \frac{y^2}{4} \geq 0
\]

This inequality defines an ellipse:

\[
x^2 + \frac{y^2}{4} \leq 1
\]

Thus, \( (x, y) \) must lie inside or on the boundary of this ellipse.

- Second term: \( \sqrt{x - y} \)

The argument inside the square root must also be non-negative:

\[
x - y \geq 0 \quad \text{or} \quad x \geq y
\]

### Step 2: Exclude Division by Zero
The denominator of the first term \( \frac{\sqrt{1 - x^2 - \frac{y^2}{4}}}{x - y} \) cannot be zero, so:

\[
x - y \neq 0 \quad \text{or} \quad x \neq y
\]

### Step 3: Combine Conditions for Domain
- The domain must satisfy the following three conditions:
  1. \( x^2 + \frac{y^2}{4} \leq 1 \) (ellipse condition),
  2. \( x \geq y \) (non-negativity of the square root),
  3. \( x \neq y \) (no division by zero).

### Step 4: Graphical Representation
The domain of the function can be visualized as a portion of the region inside the ellipse \( x^2 + \frac{y^2}{4} = 1 \), but only for values where \( x > y \), and excluding the line \( x = y \).

I can help you plot this representation if you want! Would you like me to compute and display the graph?

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Do you have any specific questions, or would you like further clarification? Here are five related questions:

1. What happens if \( x = y \) in the given function?
2. How does the condition \( x^2 + \frac{y^2}{4} \leq 1 \) affect the domain?
3. Why is \( x \geq y \) required for the second square root term?
4. What is the effect of excluding the line \( x = y \) from the domain?
5. How would the function behave near the boundary of the ellipse?

**Tip:** Always check conditions inside square root terms and denominators when determining the domain of a function!

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

Explore the domain of a multivariable function involving ellipses and square roots. Learn how to find the domain of f(x, y) = \frac{\sqrt{1 - x^2 - \frac{y^2}{4}}}{x - y} + \sqrt{x - y} by solving inequalities and analyzing the behavior of the function inside the ellipse.

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