https://r2.erweima.ai/mathbot/image/1719266538477pznefaml.png

Use the graph shown to find the following.
(a) The domain and range of the function
(b) The intercepts, if any
(c) Horizontal asymptotes, if any
(d) Vertical asymptotes, if any
(e) Oblique asymptotes, if any

To analyze the given graph and answer the questions, let's first identify key characteristics from the graph.

### (a) Domain and Range of the Function
- **Domain:** The domain includes all real numbers except where there are vertical asymptotes or points of discontinuity. In the graph, there are vertical asymptotes at \( x = -2 \) and \( x = 2 \). Thus, the domain is \( x \in (-\infty, -2) \cup (-2, 2) \cup (2, \infty) \).
- **Range:** The range includes all real numbers except where the function cannot take values. Given the vertical asymptotes at \( x = -2 \) and \( x = 2 \), and the horizontal asymptote, the range is \( y \in (-\infty, -1) \cup (-1, \infty) \).

### (b) Intercepts
- **x-intercepts:** The points where the graph crosses the x-axis. From the graph, it does not cross the x-axis, so there are no x-intercepts.
- **y-intercepts:** The points where the graph crosses the y-axis. The graph does not cross the y-axis, so there are no y-intercepts.

### (c) Horizontal Asymptotes
- The horizontal asymptote is observed as \( y = -1 \). This is where the graph approaches as \( x \) goes to \( \infty \) or \( -\infty \).

### (d) Vertical Asymptotes
- The vertical asymptotes are where the graph heads towards \( \pm \infty \) as \( x \) approaches a certain value. From the graph, these are at \( x = -2 \) and \( x = 2 \).

### (e) Oblique Asymptotes
- Oblique asymptotes occur when the graph approaches a line that is neither horizontal nor vertical as \( x \) goes to \( \infty \) or \( -\infty \). There are no oblique asymptotes in this graph.

To summarize:
- **Domain:** \( (-\infty, -2) \cup (-2, 2) \cup (2, \infty) \)
- **Range:** \( (-\infty, -1) \cup (-1, \infty) \)
- **Intercepts:** None
- **Horizontal Asymptotes:** \( y = -1 \)
- **Vertical Asymptotes:** \( x = -2 \) and \( x = 2 \)
- **Oblique Asymptotes:** None

Would you like any more details or have any questions?

Here are 5 additional questions you might consider:
1. How do you determine the horizontal asymptote of a function?
2. What are the steps to find the vertical asymptotes of a rational function?
3. Can a function have both horizontal and oblique asymptotes?
4. How do you find the domain of a function that includes a square root?
5. What are the characteristics of a rational function?

**Tip:** To find the vertical asymptotes of a rational function, set the denominator equal to zero and solve for \( x \).

Learn how to analyze a function's characteristics using its graph. Understand the domain, range, intercepts, horizontal asymptotes, vertical asymptotes, and oblique asymptotes through a detailed graphical analysis.

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