https://file.aiquickdraw.com/mathbot/file/1730081004844sspny98c.jpg

Let's analyze and solve the questions based on the given graph.

### Observations from the Graph:
1. **Asymptotes**:
   - There is a vertical asymptote at \( x = -2 \).
   - There is a horizontal asymptote at \( y = 3 \).

2. **Intercepts**:
   - The graph crosses the x-axis at \( x = -4 \).
   - The graph crosses the y-axis at \( y = 0 \).

### Solutions:

#### (a) Find all x-intercepts and y-intercepts.
- **x-intercept(s)**: The graph crosses the x-axis at \( x = -4 \), so the x-intercept is \( x = -4 \).
- **y-intercept(s)**: The graph crosses the y-axis at \( y = 0 \), so the y-intercept is \( y = 0 \).

**Answers**:
- x-intercept(s): \(-4\)
- y-intercept(s): \(0\)

#### (b) Write the equations for all vertical and horizontal asymptotes.
- **Vertical Asymptote**: The vertical asymptote is located at \( x = -2 \).
- **Horizontal Asymptote**: The horizontal asymptote is located at \( y = 3 \).

**Answers**:
- Vertical asymptote(s): \( x = -2 \)
- Horizontal asymptote(s): \( y = 3 \)

#### (c) Find the domain and range of \( f \).
- **Domain**: The function is undefined at \( x = -2 \) due to the vertical asymptote. Therefore, the domain is all real numbers except \( x = -2 \), which can be written as:
  \[
  (-\infty, -2) \cup (-2, \infty)
  \]
- **Range**: The function has a horizontal asymptote at \( y = 3 \), so it approaches \( y = 3 \) but never reaches it. The range is therefore all real values except \( y = 3 \), which can be written as:
  \[
  (-\infty, 3) \cup (3, \infty)
  \]

**Answers**:
- Domain: \( (-\infty, -2) \cup (-2, \infty) \)
- Range: \( (-\infty, 3) \cup (3, \infty) \)

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Would you like further details on any part of this solution?

Here are some related questions for practice:
1. How do you find vertical asymptotes for a rational function?
2. What are horizontal asymptotes and how are they determined?
3. Can a rational function have more than one vertical asymptote?
4. How does the degree of the polynomial in the numerator and denominator affect the horizontal asymptote?
5. What changes in the function would alter the domain and range?

**Tip:** When analyzing graphs, identifying asymptotes and intercepts can help understand the behavior of the function over its entire domain.

The graph of a rational function f is shown below. Assume that all asymptotes and intercepts are shown and that the graph has no 'holes'.
(a) Find all x-intercepts and y-intercepts. Check all that apply.
(b) Write the equations for all vertical and horizontal asymptotes.
(c) Find the domain and range of f.

Learn how to analyze the intercepts, asymptotes, domain, and range of a rational function based on its graph. This problem involves finding x- and y-intercepts, writing equations for vertical and horizontal asymptotes, and determining the domain and range. Suitable for students in Grades 10-12.

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