select the correct answer. consider the graph of the function ( f(x)=log x ). which is a feature of function…

Explanation:

Step1: Analyze the transformation of the function

The parent function is (y = \log x). The function (g(x)=-4\log(x - 8)) is a transformation of the parent function. The transformation involves a horizontal shift (8) units to the right (because of (x-8)) and a vertical stretch by a factor of (4) and a reflection about the (x) - axis (because of (- 4)).

Step2: Analyze the domain

For the function (y=\log u), the argument (u>0). For (g(x)=-4\log(x - 8)), we set (x-8>0), so (x>8). So, option D ((x < 8)) is incorrect.

Step3: Analyze the range

The range of the parent function (y = \log x) is ((-\infty,\infty)). For (y = a\log(x - h)+k) (in our case (a=-4), (h = 8), (k = 0)), the range is still ((-\infty,\infty)). So, option B ((y>-8)) is incorrect.

Step4: Analyze the end - behavior

As (x\rightarrow+\infty), for the parent function (y=\log x), (y\rightarrow+\infty). For (g(x)=-4\log(x - 8)), when (x\rightarrow+\infty), (\log(x - 8)\rightarrow+\infty), and (g(x)=-4\log(x - 8)\rightarrow-\infty). So the value of the function (g(x)) decreases as (x) approaches positive infinity. Option C (function increases as (x) approaches (+\infty)) is incorrect.

Answer:

A. The value of the function decreases as (x) approaches positive infinity.