g(x) = 3|x + 4| − 2. Which statement is correct?
	The function f (x) has the same range as g(x).
	The function f (x) has the same vertex as g(x).
	The function f (x) has the same x-intercept as g(x).
	The function f (x) has the same y-intercept as g(x).

Let's break down the given function \( g(x) = 3|x + 4| - 2 \) and evaluate the potential comparisons to a function \( f(x) \).

### Vertex of \( g(x) \):

1. **Absolute Value Function**:
   - The general form of an absolute value function is \( g(x) = a|x - h| + k \), where \( (h, k) \) is the vertex of the function.
   - In \( g(x) = 3|x + 4| - 2 \), we can rewrite the expression inside the absolute value as \( |x - (-4)| \), which means \( h = -4 \) and \( k = -2 \).
   - So, the vertex of \( g(x) \) is \( (-4, -2) \).

### X-intercept of \( g(x) \):
To find the x-intercept, set \( g(x) = 0 \):
\[
3|x + 4| - 2 = 0
\]
\[
3|x + 4| = 2
\]
\[
|x + 4| = \frac{2}{3}
\]
This gives two solutions:
\[
x + 4 = \frac{2}{3} \quad \text{or} \quad x + 4 = -\frac{2}{3}
\]
\[
x = -4 + \frac{2}{3} = -\frac{10}{3} \quad \text{or} \quad x = -4 - \frac{2}{3} = -\frac{14}{3}
\]
So, the x-intercepts of \( g(x) \) are \( x = -\frac{10}{3} \) and \( x = -\frac{14}{3} \).

### Y-intercept of \( g(x) \):
To find the y-intercept, set \( x = 0 \):
\[
g(0) = 3|0 + 4| - 2 = 3(4) - 2 = 12 - 2 = 10
\]
So, the y-intercept of \( g(x) \) is \( 10 \).

---

Now, let's evaluate the possible options:

1. **The function \( f(x) \) has the same range as \( g(x) \)**: 
   - The range of \( g(x) \) depends on the vertical stretch and shift. Since the vertex is \( (-4, -2) \), the function will have a minimum value at \( y = -2 \), and the range is \( [-2, \infty) \). If \( f(x) \) has the same range, this could be true, but more information on \( f(x) \) is needed to confirm.
   
2. **The function \( f(x) \) has the same vertex as \( g(x) \)**:
   - The vertex of \( g(x) \) is \( (-4, -2) \). If \( f(x) \) has the same vertex, this could be correct.

3. **The function \( f(x) \) has the same x-intercept as \( g(x) \)**:
   - The x-intercepts of \( g(x) \) are \( x = -\frac{10}{3} \) and \( x = -\frac{14}{3} \). More information on \( f(x) \) is needed to confirm this.

4. **The function \( f(x) \) has the same y-intercept as \( g(x) \)**:
   - The y-intercept of \( g(x) \) is \( 10 \). If \( f(x) \) also has the same y-intercept, this would be true.

---

### Conclusion:
Without more details about \( f(x) \), we cannot definitively determine which of the options is correct. However, based on common comparisons between functions like this, the **most likely correct statement** is:

**"The function \( f(x) \) has the same y-intercept as \( g(x) \)"**, assuming that \( f(x) \) shares the y-intercept of 10 with \( g(x) \).

Would you like more details or clarifications on any part of the solution? Here are some additional questions that could help expand understanding:

1. What would the range of \( f(x) \) look like if it had a different vertical shift?
2. How does the slope of \( g(x) \) change based on the coefficient of the absolute value?
3. Could \( f(x) \) have the same x-intercepts as \( g(x) \) if it's a different form of function?
4. How does changing the vertex affect the graph of absolute value functions?
5. How would the graph of \( f(x) \) compare if it had a negative leading coefficient?

**Tip**: For absolute value functions, identifying the vertex and intercepts is a key step in analyzing their graphs.

g(x) = 3|x + 4| − 2. Which statement is correct?
- The function f(x) has the same range as g(x).
- The function f(x) has the same vertex as g(x).
- The function f(x) has the same x-intercept as g(x).
- The function f(x) has the same y-intercept as g(x).

Learn how to analyze the absolute value function g(x) = 3|x + 4| - 2 by determining its vertex, x-intercept, y-intercept, and range. This detailed explanation is ideal for students in Grades 9-11 exploring advanced algebra concepts.

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