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).

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 $x$.

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 $x \to \pm \infty$.

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.