The function you provided is:

\[
f(x) = \frac{1}{x} - 3
\]

This represents a rational function. Let's break down its key features:

### 1. **Domain**:
The function is undefined where the denominator is zero. Since \( \frac{1}{x} \) has a denominator of \( x \), the function is undefined at \( x = 0 \). Therefore, the domain is all real numbers except 0:

\[
\text{Domain}: (-\infty, 0) \cup (0, \infty)
\]

### 2. **Vertical Asymptote**:
There is a vertical asymptote where the denominator equals zero, which is at \( x = 0 \).

### 3. **Horizontal Asymptote**:
As \( x \to \infty \) or \( x \to -\infty \), the term \( \frac{1}{x} \) approaches 0. Hence, the function tends towards:

\[
f(x) \approx -3
\]

So, the horizontal asymptote is \( y = -3 \).

### 4. **Intercepts**:
- **y-intercept**: There is no y-intercept since the function is undefined at \( x = 0 \).
- **x-intercept**: Set \( f(x) = 0 \) to solve for the x-intercept:
  \[
  \frac{1}{x} - 3 = 0
  \]
  Solving for \( x \):
  \[
  \frac{1}{x} = 3 \implies x = \frac{1}{3}
  \]
  Therefore, the x-intercept is \( x = \frac{1}{3} \).

### 5. **Behavior**:
- For \( x > 0 \), \( f(x) \) is positive, approaching 0 from above as \( x \to \infty \) and increasing rapidly as \( x \to 0^+ \).
- For \( x < 0 \), \( f(x) \) is negative, approaching 0 from below as \( x \to -\infty \) and decreasing rapidly as \( x \to 0^- \).

Would you like more details or have any specific questions? 

Here are 5 relative questions that might expand your understanding:
1. How do vertical and horizontal asymptotes influence the graph's shape?
2. What happens if you shift the graph of \( f(x) = \frac{1}{x} \) up or down?
3. Can we rewrite the function in a different form to reveal more characteristics?
4. What is the difference in behavior of \( f(x) \) as \( x \to \infty \) vs. \( x \to 0 \)?
5. How would you graph this function step by step?

**Tip**: When analyzing a rational function, always check for vertical asymptotes by setting the denominator equal to zero, and for horizontal asymptotes by checking the limits at \( x \to \infty \).

Explore the rational function f(x) = 1/x - 3, focusing on its vertical and horizontal asymptotes, intercepts, and domain. This solution breaks down the function’s key features, such as the behavior at x = 0 and x = 1/3. Perfect for high school students in Grades 9-12.

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