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:

$(-\infty, 2) \cup (2, \infty)$

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 $x \to -\infty$.