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Let's go through each part of question 2 for the function \( f(x) = \frac{3}{x - 2} \):

### a) Vertical Asymptote
The vertical asymptote occurs where the denominator of \( f(x) \) is zero. 

Since the denominator is \( x - 2 \), setting it equal to zero gives:
\[
x - 2 = 0 \Rightarrow x = 2
\]
**The vertical asymptote is \( x = 2 \).**

### b) Table of Values Near the Vertical Asymptote
To observe the behavior of \( f(x) \) near \( x = 2 \), let's evaluate \( f(x) \) for values approaching 2 from both sides (from the left and right).

| \( x \)   | \( f(x) = \frac{3}{x - 2} \) |
|-----------|-----------------------------|
| 1.9       | \( f(1.9) = \frac{3}{1.9 - 2} = -30 \) |
| 1.99      | \( f(1.99) = \frac{3}{1.99 - 2} = -300 \) |
| 2.01      | \( f(2.01) = \frac{3}{2.01 - 2} = 300 \) |
| 2.1       | \( f(2.1) = \frac{3}{2.1 - 2} = 30 \) |

As \( x \to 2^- \), \( f(x) \to -\infty \), and as \( x \to 2^+ \), \( f(x) \to +\infty \).

### c) Horizontal Asymptote
The horizontal asymptote is determined by analyzing the behavior of \( f(x) \) as \( x \to \pm \infty \).

\[
f(x) = \frac{3}{x - 2} \approx \frac{3}{x} \text{ as } x \to \infty \Rightarrow f(x) \to 0
\]

**The horizontal asymptote is \( y = 0 \).**

### d) Table of Values Near the Horizontal Asymptote
To confirm the end behavior, we can observe values of \( f(x) \) for large positive and negative \( x \).

| \( x \) | \( f(x) = \frac{3}{x - 2} \) |
|---------|-----------------------------|
| 10      | \( f(10) = \frac{3}{10 - 2} = 0.375 \) |
| 100     | \( f(100) = \frac{3}{100 - 2} \approx 0.0306 \) |
| -10     | \( f(-10) = \frac{3}{-10 - 2} = -0.25 \) |
| -100    | \( f(-100) = \frac{3}{-100 - 2} \approx -0.0294 \) |

As \( x \to \pm \infty \), \( f(x) \to 0 \).

### e) Domain and Range
- **Domain**: All real values of \( x \) except where the denominator is zero. So, \( x \neq 2 \).
  
  **Domain**: \( x \in (-\infty, 2) \cup (2, \infty) \)

- **Range**: The function can take any real value except for the horizontal asymptote \( y = 0 \).

  **Range**: \( y \in (-\infty, 0) \cup (0, \infty) \)

### f) Positive and Negative Intervals
To determine the intervals where \( f(x) \) is positive or negative, observe the sign of \( f(x) = \frac{3}{x - 2} \):
- For \( x < 2 \), \( f(x) \) is negative.
- For \( x > 2 \), \( f(x) \) is positive.

So:
- **Positive interval**: \( x \in (2, \infty) \)
- **Negative interval**: \( x \in (-\infty, 2) \)

### g) Sketch of the Graph
To sketch \( f(x) = \frac{3}{x - 2} \):
1. Plot the vertical asymptote at \( x = 2 \) and the horizontal asymptote at \( y = 0 \).
2. Indicate the behavior as \( x \) approaches the asymptotes.
3. Show the function approaching \( 0 \) as \( x \to \pm \infty \).

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Would you like further assistance with this problem, such as plotting the graph?

### Further Questions:
1. How does the behavior of \( f(x) \) change near the asymptotes for different values of \( k \) in \( f(x) = \frac{k}{x - a} \)?
2. What happens to the horizontal asymptote if we change \( f(x) \) to \( \frac{3x}{x - 2} \)?
3. How can transformations shift the location of the asymptotes?
4. In what cases would the function have no horizontal asymptote?
5. How would you identify any slant asymptotes for a rational function?

### Tip:
When analyzing rational functions, always start by identifying asymptotes—they provide a good framework for understanding the function's behavior.

Consider the function f(x) = 3 / (x - 2) with various tasks including determining asymptotes, creating a table of values near asymptotes, finding domain and range, identifying positive and negative intervals, and sketching the graph.

Explore the rational function f(x) = 3 / (x - 2), covering vertical and horizontal asymptotes, domain and range, intervals of positivity and negativity, and graph sketching. Suitable for high school math students in Grades 11-12.

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