https://file.aiquickdraw.com/mathbot/file/1726668923421glmx6p3u.jpg

The graph in the image shows a rational function with both a vertical and a horizontal asymptote. Let's break down the information from the graph and the question:

1. **Vertical Asymptote:** From the graph, the vertical asymptote appears to be at \( x = 0 \), as the graph sharply approaches infinity and negative infinity near this line.

2. **Horizontal Asymptote:** The graph approaches the horizontal line \( y = 2 \) as \( x \) increases or decreases without bound.

### Question 1: The domain of this function
The domain refers to the set of all possible input values \( x \) for which the function is defined. Since there is a vertical asymptote at \( x = 0 \), the function is undefined there. Therefore, the domain is all real numbers except \( x = 0 \). In interval notation, the domain is:

\[
(-\infty, 0) \cup (0, \infty)
\]

### Question 2: The range of this function
The range refers to the set of all possible output values \( y \). The graph suggests that the function approaches a horizontal asymptote at \( y = 2 \), but the function never actually reaches \( y = 2 \). The range excludes \( y = 2 \) but covers all other values of \( y \). Therefore, the range is:

\[
(-\infty, 2) \cup (2, \infty)
\]

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Let me know if you'd like further clarification on any points. 

Here are five related questions for further exploration:
1. What is the equation of a function with a vertical asymptote at \( x = 3 \) and a horizontal asymptote at \( y = -1 \)?
2. How do you calculate horizontal asymptotes for rational functions in general?
3. What are the types of asymptotes a function can have?
4. What is the significance of a vertical asymptote in the real-world applications of functions?
5. Can a function have more than one vertical or horizontal asymptote?

**Tip:** When analyzing the behavior of a rational function, remember that vertical asymptotes occur where the denominator is zero, and horizontal asymptotes describe the function's behavior as \( x \to \infty \) or \( x \to -\infty \).

A function's graph is shown with both a vertical and horizontal asymptote. The user is asked to determine the domain and range of the function using interval notation.

This problem explores how to determine the domain and range of a rational function with a vertical asymptote at x=0 and a horizontal asymptote at y=2. Using interval notation, the solution covers graph analysis and asymptote behavior. Suitable for high school students.

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