The graph you provided appears to represent a rational function, possibly of the form $f(x) = \frac{1}{x}$, where the behavior includes a vertical asymptote and horizontal asymptote. Let's analyze the domain, range, and asymptotes from the graph.

1. Domain:

• The domain represents the set of all $x$-values for which the function is defined.
• From the graph, we observe a vertical asymptote at $x = 0$, meaning the function is undefined at $x = 0$.
• Therefore, the domain excludes $x = 0$, so: $\text{Domain: } (-\infty, 0) \cup (0, \infty)$

2. Range:

• The range is the set of all possible $y$-values the function can take.
• The graph has a horizontal asymptote at $y = 0$, implying the function never reaches $y = 0$, but it approaches it as $x$ increases or decreases.
• Since the graph shows values of $y$ moving towards infinity on both sides but never touching $y = 0$, the range excludes $y = 0$, so: $\text{Range: } (0, \infty)$

3. Asymptotes:

• Vertical asymptote at $x = 0$.
• Horizontal asymptote at $y = 0$.

Summary:

• Domain: $(-\infty, 0) \cup (0, \infty)$
• Range: $(0, \infty)$
• Vertical asymptote: $x = 0$
• Horizontal asymptote: $y = 0$

Would you like more details on asymptotes or other features of this graph? Here are some related questions:

1. How do asymptotes affect the behavior of rational functions?
2. What types of discontinuities exist in functions like this one?
3. How can we determine horizontal asymptotes for more complex rational functions?
4. What would happen to the graph if the function had a different denominator (e.g., $\frac{1}{x-1}$)?
5. How does the graph change when vertical or horizontal translations are applied?

Tip: Always check for asymptotes when analyzing rational functions, as they define key limitations in the domain and range.