The question asks about the shape of the graph for $f(x) = \frac{1}{x}$.

Explanation:

The graph of $f(x) = \frac{1}{x}$ is a hyperbola. It consists of two branches:

1. One in the first quadrant, where $x > 0$ and $y > 0$,
2. One in the third quadrant, where $x < 0$ and $y < 0$.

The function has:

• A vertical asymptote at $x = 0$,
• A horizontal asymptote at $y = 0$.

This behavior is characteristic of a hyperbola, which is the correct answer.

Correct Answer: Hyperbola

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Related Questions:

1. What is an asymptote, and how does it relate to the graph of $f(x) = \frac{1}{x}$?
2. How does the graph of $f(x) = \frac{1}{x}$ differ from $f(x) = -\frac{1}{x}$?
3. How do transformations affect the graph of $f(x) = \frac{1}{x}$?
4. What other equations also form hyperbolas, and how are they derived?
5. What is the domain and range of $f(x) = \frac{1}{x}$?

Tip:

When analyzing a rational function like $\frac{1}{x}$, check for vertical and horizontal asymptotes—they can provide key insights into the graph's shape.