The problem is asking to identify the vertical and horizontal asymptotes of the given functions. Let's go through each function one by one.

Function $f(x) = \frac{x}{x - 2}$

1. Vertical Asymptote:

   • A vertical asymptote occurs where the denominator of the rational function is equal to zero, as long as the numerator does not also equal zero at the same point.
   • For $f(x) = \frac{x}{x - 2}$, the denominator is $x - 2$.
   • Setting the denominator equal to zero: $x - 2 = 0 \quad \Rightarrow \quad x = 2$
   • So, the vertical asymptote is $x = 2$.

2. Horizontal Asymptote:

   • To find the horizontal asymptote, compare the degrees of the numerator and denominator.
   • Here, the degrees of both the numerator and denominator are the same (both are degree 1).
   • When the degrees are equal, the horizontal asymptote is found by dividing the leading coefficients. The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is also 1. $y = \frac{1}{1} = 1$
   • So, the horizontal asymptote is $y = 1$.

Function $g(x) = \frac{3x - 1}{x^2 - 9}$

1. Vertical Asymptote:

   • First, factor the denominator: $x^2 - 9 = (x - 3)(x + 3)$
   • Set the denominator equal to zero: $(x - 3)(x + 3) = 0 \quad \Rightarrow \quad x = 3 \quad \text{or} \quad x = -3$
   • So, the vertical asymptotes are $x = 3$ and $x = -3$.

2. Horizontal Asymptote:

   • The degree of the numerator is 1 (since $3x - 1$ is linear), and the degree of the denominator is 2 (since $x^2 - 9$ is quadratic).
   • When the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is $y = 0$.
   • Therefore, the horizontal asymptote is $y = 0$.

Summary:

• For $f(x) = \frac{x}{x - 2}$:

  • Vertical Asymptote: $x = 2$
  • Horizontal Asymptote: $y = 1$

• For $g(x) = \frac{3x - 1}{x^2 - 9}$:

  • Vertical Asymptotes: $x = 3$, $x = -3$
  • Horizontal Asymptote: $y = 0$

Would you like further clarification or details on these steps?

Here are 5 related questions:

1. How do you find oblique (slant) asymptotes for rational functions?
2. Can a function have multiple horizontal asymptotes?
3. What happens to the behavior of a function near a vertical asymptote?
4. How can you determine the x-intercepts and y-intercepts of a rational function?
5. How do holes in rational functions differ from vertical asymptotes?

Tip: Always check if the numerator and denominator share any common factors. If they do, you may have a "hole" instead of a vertical asymptote.