This graph represents a rational function, likely of the form $f(x) = \frac{1}{x + a} + b$, which has asymptotes and undefined points.

Analyzing Domain and Range

1. Domain: The domain includes all $x$-values except where the function is undefined, usually where there is a vertical asymptote. From the graph, the vertical asymptote appears to be at $x = -2$. Therefore, the domain is: $x \in (-\infty, -2) \cup (-2, \infty)$

2. Range: The range includes all $y$-values except where there is a horizontal asymptote. The graph suggests a horizontal asymptote at $y = -1$. Thus, the range is: $y \in (-\infty, -1) \cup (-1, \infty)$

Would you like more detailed information on how to determine domains and ranges of rational functions?

Follow-Up Questions

1. How would the domain change if the vertical asymptote shifted to a different $x$-value?
2. What determines the location of the horizontal asymptote in a rational function?
3. How can you find the equation of this function based on the asymptotes and general shape?
4. What happens to the domain and range if the function is shifted up or down?
5. Can the range include the value of the horizontal asymptote in some cases?

Tip

For rational functions, remember that vertical asymptotes occur where the denominator equals zero, and horizontal asymptotes depend on the degrees of the numerator and denominator.