The given equation is:

\[
x^2 - 4y^2 + 16 = 0
\]

To find the domain and range, let's first rewrite the equation in a standard form.

### Step 1: Solve for \( x^2 \)

Subtract 16 from both sides:

\[
x^2 - 4y^2 = -16
\]

Add \( 4y^2 \) to both sides:

\[
x^2 = 4y^2 - 16
\]

### Step 2: Solve for \( x \)

Now, isolate \( x \):

\[
x^2 = 4(y^2 - 4)
\]

Taking the square root of both sides:

\[
x = \pm 2\sqrt{y^2 - 4}
\]

### Step 3: Analyze the Domain

For \( x \) to be real, the expression under the square root must be non-negative:

\[
y^2 - 4 \geq 0
\]

This simplifies to:

\[
y^2 \geq 4
\]

Thus:

\[
y \leq -2 \quad \text{or} \quad y \geq 2
\]

The domain of \( x \) depends on the values of \( y \) for which the equation is defined. Since \( y \) can take any value where \( y \leq -2 \) or \( y \geq 2 \), \( x \) is defined for these corresponding values.

### Step 4: Analyze the Range

Since \( x = \pm 2\sqrt{y^2 - 4} \), and the square root function only produces non-negative values, \( x \) will range from negative to positive values.

As \( y \) increases or decreases away from \( y = \pm 2 \), \( x \) will take larger and larger values, both positive and negative. Thus, \( x \) can take any real value, making the range of \( x \) all real numbers.

### Summary

- **Domain**: \( y \leq -2 \) or \( y \geq 2 \)
- **Range**: All real numbers \( (-\infty, \infty) \)

Would you like more details or have any other questions?

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**Related Questions:**
1. How do you determine the domain and range of an ellipse?
2. What is the domain and range of the function \( y = \sqrt{x^2 - 4} \)?
3. Can you find the intercepts of \( x^2 - 4y^2 + 16 = 0 \)?
4. How would the domain and range change if the equation was \( x^2 + 4y^2 = 16 \)?
5. What are the asymptotes of the hyperbola formed by this equation?

**Tip:** When dealing with square roots, always ensure the expression under the root is non-negative to determine the domain.

Explore the domain and range of the equation x^2 - 4y^2 + 16 = 0. Learn how to determine the valid values for x and the entire range of x based on the values of y. Detailed step-by-step analysis and solutions provided.

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