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 $x \to \infty$.