select the correct answer.\nwhich statement describes the end - behavior of the function (f(x)=3|x…

select the correct answer.\nwhich statement describes the end - behavior of the function (f(x)=3|x - 7|-7)?\na. as (x) approaches negative infinity, (f(x)) approaches negative infinity.\nb. as (x) approaches negative infinity, (f(x)) approaches positive infinity.\nc. as (x) approaches positive infinity, (f(x)) approaches negative infinity.\nd. as (x) approaches positive infinity, (f(x)) is no longer continuous.

select the correct answer.\nwhich statement describes the end - behavior of the function (f(x)=3|x - 7|-7)?\na. as (x) approaches negative infinity, (f(x)) approaches negative infinity.\nb. as (x) approaches negative infinity, (f(x)) approaches positive infinity.\nc. as (x) approaches positive infinity, (f(x)) approaches negative infinity.\nd. as (x) approaches positive infinity, (f(x)) is no longer continuous.

Answer

Explanation:

Step1: Analyze absolute - value function

The function is (f(x)=3|x - 7|-7). The absolute - value function (y = |x|) is defined as (y=\begin{cases}x, & x\geq0\-x, & x<0\end{cases}). For (y = |x - 7|), when (x\geq7), (y=x - 7); when (x<7), (y=-(x - 7)=7 - x).

Step2: Consider (x\to-\infty)

When (x\to-\infty), (|x - 7|=-(x - 7)=7 - x). Then (f(x)=3(7 - x)-7=21-3x - 7=-3x + 14). As (x\to-\infty), (-3x\to+\infty), so (f(x)\to+\infty).

Step3: Consider (x\to+\infty)

When (x\to+\infty), (|x - 7|=x - 7). Then (f(x)=3(x - 7)-7=3x-21 - 7=3x - 28). As (x\to+\infty), (3x\to+\infty), so (f(x)\to+\infty). Also, the absolute - value function (y = 3|x - 7|-7) is continuous for all real (x) since it is a composition of linear and absolute - value functions.

Answer:

B. As (x) approaches negative infinity, (f(x)) approaches positive infinity.