Question: Select the correct answer.
Which statement describes the end behavior of the function $f(x) = 3|x - 7| - 7$?
As $x$ approaches negative infinity, $f(x)$ approaches negative infinity.
As $x$ approaches negative infinity, $f(x)$ approaches positive infinity.
As $x$ approaches positive infinity, $f(x)$ approaches negative infinity.
As $x$ approaches positive infinity, $f(x)$ is no longer continuous.
1. **State the problem:** We need to determine the end behavior of the function $$f(x) = 3|x - 7| - 7$$ as $x$ approaches positive and negative infinity.
2. **Recall the properties of absolute value functions:** The absolute value function $|x|$ outputs the distance of $x$ from zero and is always non-negative. As $x \to \pm \infty$, $|x| \to \infty$.
3. **Analyze the function:**
$$f(x) = 3|x - 7| - 7$$
As $x \to \infty$, $|x - 7| \approx x - 7$ because $x - 7$ is positive and large.
So,
$$f(x) \approx 3(x - 7) - 7 = 3x - 21 - 7 = 3x - 28$$
As $x \to \infty$, $3x - 28 \to \infty$.
4. As $x \to -\infty$, $x - 7$ is very negative, so
$$|x - 7| = -(x - 7) = -x + 7$$
Thus,
$$f(x) = 3(-x + 7) - 7 = -3x + 21 - 7 = -3x + 14$$
As $x \to -\infty$, $-3x + 14 \to \infty$ because $-3x$ becomes very large positive.
5. **Conclusion:**
- As $x \to -\infty$, $f(x) \to \infty$ (positive infinity).
- As $x \to \infty$, $f(x) \to \infty$ (positive infinity).
6. **Answer choice:** The correct statement is: "As $x$ approaches negative infinity, $f(x)$ approaches positive infinity."