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 El dominio de la función
f(x,y)2x+3y+z4−x2−y2−z2
es
Pregunta 2Respuesta

a.
x2+y2+z2≥0

b.
x2+y2+z2+4>0

c.
x2+y2+z2≠4

d.
x2+y2+z2≤4

We are given the function:

\[
f(x, y, z) = \frac{2x + 3y + z}{4 - x^2 - y^2 - z^2}
\]

### Step 1: Identify Domain Constraints
To determine the domain, we must ensure that the function is defined for the given inputs. The main concern is the denominator:

\[
4 - x^2 - y^2 - z^2
\]

For the function to be defined, the denominator must **not** be equal to zero and must remain positive. Hence, we have two conditions:

1. **Denominator ≠ 0**:
   \[
   4 - x^2 - y^2 - z^2 \neq 0 \implies x^2 + y^2 + z^2 \neq 4
   \]

2. **Denominator > 0**:
   \[
   4 - (x^2 + y^2 + z^2) > 0 \implies x^2 + y^2 + z^2 < 4
   \]

### Step 2: Combine the Constraints
The domain of the function is determined by the inequality:

\[
x^2 + y^2 + z^2 < 4
\]

Thus, the expression \(x^2 + y^2 + z^2\) must be strictly less than 4 for the function to be defined.

### Step 3: Match with Options
Among the provided options:

- **a.** \(x^2 + y^2 + z^2 \geq 0\) — This is true for all real values, but does not account for the upper bound.
- **b.** \(x^2 + y^2 + z^2 + 4 > 0\) — This is irrelevant to the condition we derived.
- **c.** \(x^2 + y^2 + z^2 \neq 4\) — This is a necessary condition, but it alone does not specify the strict upper bound.
- **d.** \(x^2 + y^2 + z^2 \leq 4\) — This includes equality, but we need \(x^2 + y^2 + z^2 < 4\).

Thus, **none of the provided options directly match** the derived domain \(x^2 + y^2 + z^2 < 4\). However, option **c** partially addresses the exclusion of the value 4, though it is not sufficient alone.

Let me know if you would like further clarification or a more detailed explanation!

Here are 5 related questions to explore the topic:

1. What happens if the value of \(x^2 + y^2 + z^2\) equals 4?
2. How would the domain change if the denominator had a positive constant instead of 4?
3. Can you rewrite the function as a polar equation in spherical coordinates?
4. What is the range of values for the function \(f(x, y, z)\) given the domain constraint?
5. How would the domain change if the numerator had a different expression?

**Tip:** When analyzing the domain of a function, always focus on constraints that make the denominator non-zero and ensure that any square roots or logarithms inside the function are properly handled.

El dominio de la función f(x,y) = (2x + 3y + z) / (4 − x^2 − y^2 − z^2) es:

Explore how to find the domain of the multivariable function f(x,y) = (2x + 3y + z) / (4 − x^2 − y^2 − z^2). Understand the conditions required for the denominator to be positive and non-zero, ensuring that the function is defined. Suitable for Grades 11-12 students interested in advanced algebra and multivariable calculus.

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