The function $f(x) = \frac{1}{x}$ is a rational function, where the numerator is 1 and the denominator is $x$. This function is defined for all real values of $x$ except $x = 0$, since division by zero is undefined.

Key Characteristics of $f(x) = \frac{1}{x}$:

1. Domain: The domain is $\mathbb{R} \setminus \{0\}$, meaning all real numbers except $x = 0$.
2. Range: The range is also $\mathbb{R} \setminus \{0\}$, meaning the output can be any real number except zero.
3. Vertical Asymptote: There is a vertical asymptote at $x = 0$ because the function approaches infinity as $x$ approaches zero from either direction.
4. Horizontal Asymptote: The function has a horizontal asymptote at $y = 0$ as $x$ approaches infinity (or negative infinity).
5. Behavior:
   • As $x \to 0^+$, $f(x) \to \infty$
   • As $x \to 0^-$, $f(x) \to -\infty$
   • As $x \to \infty$ or $x \to -\infty$, $f(x) \to 0$

Would you like me to analyze any specific properties of this function, or would you like help with a related problem?

Here are some follow-up questions for you:

1. How do the asymptotes affect the graph of $f(x) = \frac{1}{x}$?
2. What is the limit of $f(x) = \frac{1}{x}$ as $x \to 0^+$?
3. Can you find the derivative of $f(x) = \frac{1}{x}$?
4. What happens to $f(x) = \frac{1}{x}$ as $x \to \infty$?
5. How do transformations like shifts and stretches affect the graph of $f(x) = \frac{1}{x}$?

Tip: When working with rational functions, it’s helpful to identify the domain, asymptotes, and behavior at critical points like zeros or infinities to understand the overall shape of the graph.