https://file.aiquickdraw.com/mathbot/file/1724735714209ekkxqni6.jpg

Based on the graph provided, we can analyze the domain and range of the rational function.

### **Domain:**
The domain of a function is the set of all possible input values (x-values) that the function can accept. For rational functions, the domain is typically all real numbers except where the function has vertical asymptotes (which occur where the denominator is zero).

From the graph:
- There is a vertical asymptote at \( x = 2 \).

This means that the function is undefined at \( x = 2 \), so the domain excludes this value.

**Domain:** \( (-\infty, 2) \cup (2, \infty) \)

### **Range:**
The range is the set of all possible output values (y-values) of the function.

From the graph:
- There is a horizontal asymptote at \( y = 0 \), indicating that the function never actually reaches \( y = 0 \), but it approaches it as \( x \) becomes very large or very small.
- The graph extends infinitely upwards and downwards as \( x \) approaches the vertical asymptote from the left and right sides.

**Range:** \( (-\infty, 0) \cup (0, \infty) \)

### Summary:
- **Domain:** \( (-\infty, 2) \cup (2, \infty) \)
- **Range:** \( (-\infty, 0) \cup (0, \infty) \)

Would you like more details or have any questions about this? Here are five related questions to deepen your understanding:

1. How do you find vertical asymptotes of a rational function algebraically?
2. What is the significance of horizontal asymptotes in determining the range of a function?
3. How does the graph of a rational function differ when there is a hole instead of an asymptote?
4. What are the steps to find the x-intercepts and y-intercepts of a rational function?
5. Can a rational function have more than one vertical asymptote?

**Tip:** When analyzing the domain and range from a graph, always check for asymptotes and holes, as these indicate where the function is undefined or restricted.

Explore the domain and range of rational functions through detailed analysis and examples. Learn how to identify vertical asymptotes and horizontal asymptotes, crucial for understanding where the function is defined and its output values. Suitable for Grades 11-12 students seeking a deeper understanding of function behavior.

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