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 $10$.

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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.

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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.