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The problem involves graphing the rational function \( F(x) = 9 + \frac{1}{x} \), finding the domain and range from the graph, and identifying any asymptotes.

### (a) Graphing the function \( F(x) = 9 + \frac{1}{x} \) using transformations:
The base function is \( \frac{1}{x} \), and the transformation involves shifting the graph vertically by 9 units (since the "+9" is added). The resulting graph will have the following properties:
- The vertical asymptote will be at \( x = 0 \) (same as for \( \frac{1}{x} \)).
- The horizontal asymptote will shift from \( y = 0 \) to \( y = 9 \) due to the vertical translation.

### (b) Domain and Range:
- **Domain**: The function is undefined at \( x = 0 \), so the domain is \( (-\infty, 0) \cup (0, \infty) \).
- **Range**: The horizontal asymptote is at \( y = 9 \), meaning the function will approach but never reach \( y = 9 \). Therefore, the range is \( (-\infty, 9) \cup (9, \infty) \).

### (c) Asymptotes:
- **Vertical Asymptote**: \( x = 0 \)
- **Horizontal Asymptote**: \( y = 9 \)

### Correct graph:
From the images provided, the correct graph should show a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = 9 \), with the general shape of \( \frac{1}{x} \) shifted upward. Based on the images:
- Graph C is the correct one, as it shows both the vertical and horizontal asymptotes in the correct positions and the graph behavior matches the transformations described.

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Do you need further details or explanations?

Here are five related questions for you to explore:
1. How would the graph change if the transformation was \( F(x) = 9 - \frac{1}{x} \)?
2. How do you find the horizontal asymptote for a rational function with a different form?
3. Can a rational function have more than one vertical asymptote? If so, how?
4. How does the vertical stretch affect the graph of a rational function?
5. What are the steps to graph a rational function without a graphing calculator?

**Tip**: Always identify asymptotes first when graphing rational functions. They help guide the general shape of the graph.

For the function F(x) = 9 + 1/x, (a) graph the rational function using transformations, (b) use the final graph to find the domain and range, and (c) use the final graph to list any vertical, horizontal, or oblique asymptotes.

Learn how to graph the rational function F(x) = 9 + 1/x using transformations. This detailed solution covers identifying vertical and horizontal asymptotes, determining domain and range, and graphing techniques. Suitable for Grades 10-12.

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