Question

1. examine the function ( f(x)=2-log _{2}(x - 3) ). which statement does not accurately describe the given function? a. the domain of ( f(x) ) is ( (3,infty) ) b. the graph of ( f(x) ) has a horizontal asymptote at ( y = 2 ) c. the graph of ( f(x) ) has a vertical asymptote at ( x = 3 ) d. ( f(7)=0 )

Explanation:

Step1: Analyze the domain

For the function \(y = \log_{2}(x - 3)\), the argument of the logarithm \(x-3>0\), so \(x>3\). The domain of \(f(x)=2-\log_{2}(x - 3)\) is \((3,\infty)\). So option A is correct.

Step2: Analyze the vertical asymptote

As \(x
ightarrow3^{+}\), \(\log_{2}(x - 3)
ightarrow-\infty\), and \(y = 2-\log_{2}(x - 3)
ightarrow+\infty\). The vertical asymptote is \(x = 3\). So option C is correct.

Step3: Analyze \(f(7)\)

Substitute \(x = 7\) into \(f(x)\): \(f(7)=2-\log_{2}(7 - 3)=2-\log_{2}(4)\). Since \(\log_{2}(4)=\log_{2}(2^{2}) = 2\), then \(f(7)=2 - 2=0\). So option D is correct.

Step4: Analyze the horizontal asymptote

The function \(y=\log_{2}(x-3)\) has no horizontal asymptote. For \(y = 2-\log_{2}(x - 3)\), as \(x
ightarrow+\infty\), \(\log_{2}(x - 3)
ightarrow+\infty\) and \(y=2-\log_{2}(x - 3)
ightarrow-\infty\). There is no horizontal asymptote at \(y = 2\).

Answer:

B. The graph of \(f(x)\) has a horizontal asymptote at \(y = 2\)