Let's determine the domain and range for both the linear function and the rational function (hyperbola) depicted in the graph.

### 1. **Linear Function:**

- **Domain:** The domain of a linear function is all real numbers unless there are specific restrictions indicated by the graph. In this case, the line appears to continue infinitely in both directions, so the domain is:
  \[
  \text{Domain: } (-\infty, \infty)
  \]

- **Range:** Similarly, the range of the linear function is also all real numbers, since the line covers all possible \(y\)-values as it extends infinitely:
  \[
  \text{Range: } (-\infty, \infty)
  \]

### 2. **Rational Function:**

- **Domain:** The rational function (hyperbola) shown has a vertical asymptote at \(x = 0\), meaning the function is undefined at this point. Additionally, there is an open circle at \((-4, -2)\), indicating the function is not defined at \(x = -4\) as well. Therefore, the domain is all real numbers except \(x = 0\) and \(x = -4\):
  \[
  \text{Domain: } (-\infty, -4) \cup (-4, 0) \cup (0, \infty)
  \]

- **Range:** The function has a horizontal asymptote at \(y = 0\), meaning \(y\) approaches but never actually reaches 0. Moreover, there is a gap at \(y = -2\) corresponding to the open circle at \((-4, -2)\). Thus, the range is all real numbers except \(y = 0\) and \(y = -2\):
  \[
  \text{Range: } (-\infty, -2) \cup (-2, 0) \cup (0, \infty)
  \]

### Summary:
- **Linear Function:**
  - Domain: \((-∞, ∞)\)
  - Range: \((-∞, ∞)\)
  
- **Rational Function:**
  - Domain: \((-∞, -4) \cup (-4, 0) \cup (0, ∞)\)
  - Range: \((-∞, -2) \cup (-2, 0) \cup (0, ∞)\)

Would you like a deeper analysis or have any questions about this? Here are some related questions:

1. How does the presence of asymptotes influence the range of a function?
2. What are the methods to determine the domain and range for more complex functions?
3. How can open circles on a graph affect the domain and range?
4. What are other examples of functions with restricted domains or ranges?
5. How would you find the inverse of a function, and how does this affect its domain and range?

**Tip:** When analyzing the domain and range, always consider discontinuities like holes or asymptotes, as they can create exceptions in otherwise continuous intervals.

Explore how to determine the domain and range of linear and rational functions with detailed explanations and examples. Learn about asymptotes, discontinuities, and their impact on function behavior. Suitable for high school students studying functions in mathematics.

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