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

a) State the equation of the vertical asymptote.

To find the vertical asymptote, identify the values of $x$ for which the function is undefined, i.e., where the denominator is zero.

$x - 2 = 0 \Rightarrow x = 2$

Thus, the vertical asymptote is at $x = 2$.

b) Use a table of values to determine the behaviour(s) of the function near its vertical asymptote.

We will observe the behavior of $f(x)$ as $x$ approaches 2 from the left and the right. Here’s a table of values near $x = 2$.

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

c) State the equation of the horizontal asymptote.

For rational functions, the horizontal asymptote is determined by comparing the degrees of the numerator and the denominator.

Since the numerator is a constant (degree 0) and the denominator is a first-degree polynomial (degree 1), the horizontal asymptote is $y = 0$.

d) Use a table of values to determine the end behaviours of the function near its horizontal asymptote.

We’ll examine the behavior of $f(x)$ as $x$ approaches large positive and large negative values.

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

e) Determine the domain and range.

• Domain: All real values of $x$ except $x = 2$, where the function is undefined. So, the domain is $x \in (-\infty, 2) \cup (2, \infty)$.
• Range: The function can take any real value except 0 (due to the horizontal asymptote). Thus, the range is $y \in (-\infty, 0) \cup (0, \infty)$.

f) Determine the positive and negative intervals.

• Positive Interval: $f(x) > 0$ for $x < 2$.
• Negative Interval: $f(x) < 0$ for $x > 2$.

g) Sketch the graph.

The graph of $f(x) = \frac{3}{x - 2}$ will have:

• A vertical asymptote at $x = 2$.
• A horizontal asymptote at $y = 0$.
• The function approaches $+\infty$ as $x \to 2^-$ and $-\infty$ as $x \to 2^+$.
• The graph approaches zero as $x \to \pm \infty$.

Would you like further details on any specific part? Here are five questions that might expand your understanding:

1. How do we determine the asymptotes for other types of rational functions?
2. Why does the function change signs around the vertical asymptote?
3. How would the graph change if the numerator were different?
4. What are some real-world applications of rational functions with vertical asymptotes?
5. How does one interpret the domain and range in practical terms?

Tip: Always check for vertical asymptotes by setting the denominator to zero in rational functions.