Let's analyze the given function and determine the correct asymptotes based on the options provided.

We are given the function:

$f(x) = \frac{3x^2 - 3x}{x^2 - 1}$

Step 1: Simplify the function

We can factor both the numerator and the denominator:

$f(x) = \frac{3x(x - 1)}{(x - 1)(x + 1)}$

Notice that $(x - 1)$ appears in both the numerator and the denominator, so we can cancel it out, but we must remember that $x = 1$ is a point of discontinuity.

Thus, the simplified function is:

$f(x) = \frac{3x}{x + 1}, \quad x \neq 1$

Step 2: Analyze Vertical Asymptotes (Asimtot Tegak)

Vertical asymptotes occur where the denominator is zero, but the numerator is not zero. In this case, the simplified denominator $x + 1 = 0$ gives $x = -1$.

So, there is a vertical asymptote at $x = -1$.

Additionally, since $x = 1$ was canceled during simplification, $x = 1$ is a removable discontinuity, not a vertical asymptote.

Step 3: Analyze Horizontal Asymptotes (Asimtot Datar)

To find horizontal asymptotes, we examine the behavior of $f(x)$ as $x \to \infty$ or $x \to -\infty$. For large values of $x$, the highest degree terms dominate:

$f(x) = \frac{3x}{x + 1} \approx \frac{3x}{x} = 3 \quad \text{as} \quad x \to \infty$

Thus, the horizontal asymptote is $y = 3$.

Step 4: Conclusion on the Statements

Now let's evaluate each statement:

1. Garis $x = 1$ merupakan asimtot tegak fungsi $f(x)$:

   • This is false because $x = 1$ is a removable discontinuity, not a vertical asymptote.

2. Garis $x = -1$ merupakan asimtot tegak fungsi $f(x)$:

   • This is true because $x = -1$ is a vertical asymptote.

3. Garis $x = 3$ merupakan asimtot tegak fungsi $f(x)$:

   • This is false because $x = 3$ is not an asymptote.

4. Garis $y = 3$ merupakan asimtot datar fungsi $f(x)$:

   • This is true because the horizontal asymptote is $y = 3$.

5. Garis $y = 0$ merupakan asimtot datar fungsi $f(x)$:

   • This is false because the horizontal asymptote is $y = 3$, not $y = 0$.

Final Answer:

The correct statements are 2 and 4.

Do you have any questions or need further details?

Here are 5 related questions for deeper understanding:

1. How do we distinguish between a removable discontinuity and a vertical asymptote?
2. What happens to the function near a vertical asymptote?
3. Can a function have more than one horizontal asymptote? If so, in what cases?
4. How can we apply limits to find horizontal asymptotes for rational functions?
5. What are slant (oblique) asymptotes, and when do they occur?

Tip: When simplifying rational functions, always check for removable discontinuities by factoring both the numerator and the denominator!