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The image contains three graphs with the prompt asking which of the following best represents the graph of the function \( f(x) = \frac{1}{x} - 1 \) for the interval \( [-2, 2] \).

Let’s first analyze the function \( f(x) = \frac{1}{x} - 1 \).

### Key Features of \( f(x) = \frac{1}{x} - 1 \):
1. **Asymptotes**:
   - There is a vertical asymptote at \( x = 0 \), because \( \frac{1}{x} \) is undefined at \( x = 0 \).
   - The horizontal asymptote is \( y = -1 \), because as \( x \to \infty \) or \( x \to -\infty \), the \( \frac{1}{x} \) term approaches 0, leaving \( y = -1 \).

2. **Behavior in Intervals**:
   - For \( x > 0 \) (positive values of \( x \)), \( \frac{1}{x} \) is positive, and the graph is above \( y = -1 \).
   - For \( x < 0 \) (negative values of \( x \)), \( \frac{1}{x} \) is negative, and the graph will be below \( y = -1 \).
   
3. **Near the Vertical Asymptote ( \( x = 0 \)**):
   - As \( x \to 0^+ \) (from the right), \( f(x) \to \infty \).
   - As \( x \to 0^- \) (from the left), \( f(x) \to -\infty \).

Now, let's compare this with the given graphs:

- **Graph A**: This graph shows a function that passes through the origin and resembles an \( S \)-shaped curve. This doesn't match the behavior of \( f(x) = \frac{1}{x} - 1 \) since \( f(x) \) has an asymptote at \( x = 0 \), and the curve should not pass through the origin.

- **Graph B**: This graph shows a parabolic shape, which suggests the graph of a quadratic function. This doesn't match \( f(x) = \frac{1}{x} - 1 \), which has asymptotic behavior.

- **Graph C**: This graph shows a hyperbolic shape with a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = -1 \). This is consistent with the behavior of \( f(x) = \frac{1}{x} - 1 \), making it the correct choice.

### Conclusion:
Graph **C** best represents the graph of \( f(x) = \frac{1}{x} - 1 \) for the interval \( [-2, 2] \).

Would you like more details, or do you have any other questions?

Here are 5 related questions to explore further:
1. What happens to the function \( f(x) = \frac{1}{x} - 1 \) as \( x \to \infty \)?
2. How do horizontal and vertical asymptotes affect the graph of a function?
3. What is the domain and range of \( f(x) = \frac{1}{x} - 1 \)?
4. How would the graph change if the function was \( f(x) = \frac{1}{x} + 1 \)?
5. What is the behavior of the function \( f(x) = \frac{1}{x} \) around \( x = 0 \) without any vertical shift?

**Tip**: Always check for asymptotes when dealing with rational functions, as they provide essential information about the graph's behavior.

Which of the following best represents the graph of f(x) = \frac{1}{x} - 1 for the interval [-2, 2]?

Analyze the graph of the function f(x) = 1/x - 1 within the interval [-2, 2]. Learn about vertical and horizontal asymptotes and how they affect the function's behavior. Suitable for high school students in Grades 10-12.

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