5. examine the function ( f(x)=2-log _{2}(x - 3) ). which statement does not accurately describe the given…

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\rightarrow3^{+}), (\log_{2}(x - 3)\rightarrow-\infty), and (y = 2-\log_{2}(x - 3)\rightarrow+\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\rightarrow+\infty), (\log_{2}(x - 3)\rightarrow+\infty) and (y=2-\log_{2}(x - 3)\rightarrow-\infty). There is no horizontal asymptote at (y = 2).

Answer:

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