Question

select the correct answer. consider the graph of the function ( f(x)=log x ). which is a feature of function ( g ) if ( g(x)=-4 log (x - 8) )? a. the value of the function decreases as ( x ) approaches positive infinity. b. the range is ( y>-8 ). c. the value of the function increases as ( x ) approaches positive infinity. d. the domain is ( x<8 ).

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 \(x
ightarrow+\infty\), for the parent function \(y=\log x\), \(y
ightarrow+\infty\). For \(g(x)=-4\log(x - 8)\), when \(x
ightarrow+\infty\), \(\log(x - 8)
ightarrow+\infty\), and \(g(x)=-4\log(x - 8)
ightarrow-\infty\). 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.