To find the domain and range of the equation \( xy^2 = 16 \), we analyze the equation step by step.

### 1. **Domain**:
The domain refers to all possible values of \( x \) for which the equation is defined. 

In the equation \( xy^2 = 16 \):

- \( y^2 \) is always non-negative (i.e., \( y^2 \geq 0 \)) for any real \( y \).
- To solve for \( x \), we can rewrite the equation as \( x = \frac{16}{y^2} \).
- For \( x \) to be real and defined, \( y^2 \) must not be zero (i.e., \( y \neq 0 \)) because division by zero is undefined.

Thus, the domain of \( x \) is all real numbers, except where \( y = 0 \). However, since \( y \) cannot be zero, this implies that the domain for \( x \) is all real numbers \( x \) where \( y \neq 0 \).

In terms of the \( x \) values, the domain is all real numbers \( x \neq 0 \).

### 2. **Range**:
The range refers to all possible values of \( y \) that satisfy the equation for some \( x \).

- From \( x = \frac{16}{y^2} \), we can see that \( y^2 = \frac{16}{x} \).
- Since \( y^2 \) must be non-negative, \( x \) must also be positive (i.e., \( x > 0 \)).
- For any positive \( x \), \( y^2 \) can take any positive value, meaning \( y \) can take any positive or negative value (since \( y \) is squared).

Thus, the range of \( y \) is all real numbers \( y \neq 0 \).

### **Summary**:
- **Domain**: \( x \in (-\infty, 0) \cup (0, \infty) \)
- **Range**: \( y \in (-\infty, 0) \cup (0, \infty) \)

Would you like more details or have any questions?

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Here are 5 related questions you might find interesting:
1. What is the graph of the equation \( xy^2 = 16 \) like?
2. How do the domain and range change if the equation is \( xy^3 = 16 \)?
3. What are the intercepts of the equation \( xy^2 = 16 \)?
4. How do we find the domain and range of more complex equations?
5. Can we find the inverse of the function defined by \( xy^2 = 16 \)?

**Tip:** When finding the domain and range of an equation, always consider restrictions like division by zero or taking the square root of a negative number.

Explore the domain and range of the equation xy^2 = 16 with a detailed explanation. Learn how to determine where the equation is defined and the possible values it can take. Suitable for advanced high school students.

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