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.