To solve this problem, let's analyze the given function and check for vertical and horizontal asymptotes.

Given Function

The function is: $f(x) = \frac{3x^2 - 6x}{(4 + x) |x|}$

Step 1: Vertical Asymptotes

Vertical asymptotes occur where the denominator of $f(x)$ is zero, provided that the numerator is non-zero at these points.

The denominator is: $(4 + x) |x|$ The denominator becomes zero when:

1. $x = 0$ (because $|x| = 0$ when $x = 0$),
2. $x = -4$ (since $4 + x = 0$ when $x = -4$).

Thus, the possible vertical asymptotes are at $x = 0$ and $x = -4$.

Vertical Asymptote Conclusions:

• Statement 1 (Garis $x = 0$ adalah asimtot tegak) is correct.
• Statement 3 (Garis $x = -4$ adalah asimtot tegak) is correct.

Step 2: Horizontal Asymptotes

Horizontal asymptotes are determined by the behavior of $f(x)$ as $x \to \infty$ or $x \to -\infty$.

1. For $x \to \infty$ or $x \to -\infty$, let's analyze the highest powers in the numerator and denominator:
   • The numerator $3x^2 - 6x$ has a degree of 2.
   • The denominator $(4 + x) |x|$ also behaves like $x^2$ because $|x| \cdot x$ will approach $x^2$ as $x \to \infty$ or $x \to -\infty$.

Therefore, the horizontal asymptote is given by the ratio of the leading coefficients of the highest degree terms: $y = \frac{3}{1} = 3$ So, there is a horizontal asymptote at $y = 3$.

Horizontal Asymptote Conclusions:

• Statement 4 (Garis $y = 3$ adalah asimtot datar) is correct.

Step 3: Checking Other Statements

• Statement 2 (Garis $x = 4$ adalah asimtot tegak) is incorrect because $x = 4$ does not make the denominator zero.
• Statement 5 (Garis $y = -3$ adalah asimtot datar) is incorrect because the only horizontal asymptote is at $y = 3$.

Answer

The correct statements are:

• Statement 3 (Garis $x = -4$ adalah asimtot tegak).
• Statement 4 (Garis $y = 3$ adalah asimtot datar).

Therefore, the correct answer is: $\text{b. Pernyataan 3 dan 4}$

Would you like more details on any part of this solution?

Related Questions

1. How do you determine the horizontal asymptotes for a rational function?
2. What happens to asymptotes when the highest power terms in the numerator and denominator are equal?
3. Can a function have more than one horizontal asymptote?
4. How can you determine vertical asymptotes when the denominator has absolute values?
5. Why does $x = 4$ not result in a vertical asymptote in this function?

Tip

To find vertical asymptotes, always set the denominator equal to zero and ensure the numerator is non-zero at those points.