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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?

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### 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.

Diberikan sebuah fungsi f(x) = (3x^2 - 6x) / ((4 + x) |x|). Tentukan semua pernyataan yang benar mengenai asimtot tegak dan datar fungsi f(x):
1. Garis x = 0 merupakan asimtot tegak.
2. Garis x = 4 merupakan asimtot tegak.
3. Garis x = -4 merupakan asimtot tegak.
4. Garis y = 3 merupakan asimtot datar.
5. Garis y = -3 merupakan asimtot datar.

Learn how to determine the vertical and horizontal asymptotes of the rational function f(x) = (3x^2 - 6x) / ((4 + x) |x|). This guide walks through key calculus concepts such as asymptotes, analyzing limits, and working with absolute values. Suitable for advanced high school students in Grades 11-12.

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