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Asymptotes, domain, range and x and y intercepts

Let's analyze the graph of the rational function based on the image you uploaded.

### 1. **Vertical and Horizontal Asymptotes**
   - **Vertical Asymptotes:** These occur where the function is undefined, typically where the denominator of a rational function equals zero.
     From the graph, the vertical dashed lines seem to be at \( x = -2 \) and \( x = 2 \), indicating vertical asymptotes.
     So, the equations of the vertical asymptotes are:
     \[
     x = -2 \quad \text{and} \quad x = 2
     \]

   - **Horizontal Asymptote:** This describes the end behavior of the function as \( x \to \infty \) or \( x \to -\infty \).
     The horizontal asymptote is at \( y = 0 \), as the curve approaches the x-axis but never crosses it.
     Therefore, the equation of the horizontal asymptote is:
     \[
     y = 0
     \]

### 2. **Intercepts**
   - **x-Intercept(s):** The x-intercept(s) occur where the graph crosses the x-axis (i.e., where \( y = 0 \)).
     From the graph, the curve crosses the x-axis at \( x = 0 \), meaning the x-intercept is:
     \[
     (0, 0)
     \]

   - **y-Intercept(s):** The y-intercept is where the graph crosses the y-axis (i.e., where \( x = 0 \)).
     In this case, the y-intercept also occurs at \( (0, 0) \), so it shares the same point as the x-intercept:
     \[
     (0, 0)
     \]

### 3. **Domain**
   The domain of a rational function is all real numbers except where the function is undefined (at the vertical asymptotes).
   From the graph, the function is undefined at \( x = -2 \) and \( x = 2 \). Therefore, the domain is:
   \[
   (-\infty, -2) \cup (-2, 2) \cup (2, \infty)
   \]

### 4. **Range**
   The range is the set of possible values for \( y \), excluding any values the function cannot take. Since the horizontal asymptote is at \( y = 0 \), and the graph approaches this asymptote but never touches it, the range is all real numbers except \( y = 0 \):
   \[
   (-\infty, 0) \cup (0, \infty)
   \]

Would you like more details on any part of this analysis?

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Here are five related questions you might find helpful:
1. What is the behavior of the function near the vertical asymptotes?
2. How does the horizontal asymptote affect the end behavior of the function?
3. Can you find the function's equation based on the given graph?
4. How do holes in the graph of a rational function affect the domain and range?
5. What are the different types of asymptotes and how are they determined?

### Tip: 
For rational functions, vertical asymptotes correspond to values of \( x \) that make the denominator zero, while horizontal asymptotes are determined by the degrees of the numerator and denominator.

Asymptotes, domain, range, and x and y intercepts of the rational function

Learn how to find the vertical and horizontal asymptotes, domain, range, and intercepts of a rational function. This analysis uses graph interpretation to identify key points such as x-intercepts, y-intercepts, and the behavior near asymptotes. Suitable for high school students in Grades 10-12.

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