Question

1. examine the function $f(x)=2 - \log_2(x - 3)$. which statement does not accurately describe the given function?

the domain of $f(x)$ is $(3,\infty)$
$f(7)=0$
the graph of $f(x)$ has a horizontal asymptote at $y = 2$
the graph of $f(x)$ has a vertical asymptote at $x = 3$

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 is \((3,\infty)\).

Step2: Calculate \(f(7)\)

Substitute \(x = 7\) into \(f(x)=2-\log_{2}(x - 3)\). Then \(f(7)=2-\log_{2}(7 - 3)=2-\log_{2}4=2 - 2=0\).

Step3: Analyze the vertical asymptote

For \(y=\log_{2}(x - 3)\), as \(x
ightarrow3^{+}\), \(\log_{2}(x - 3)
ightarrow-\infty\), so \(y = 2-\log_{2}(x - 3)\) has a vertical asymptote at \(x = 3\).

Step4: Analyze the horizontal asymptote

Since the function \(y=\log_{2}(x - 3)\) has no horizontal asymptote (because \(\lim_{x
ightarrow\infty}\log_{2}(x - 3)=\infty\) and \(\lim_{x
ightarrow3^{+}}\log_{2}(x - 3)=-\infty\)), \(y = 2-\log_{2}(x - 3)\) has no horizontal asymptote.

Answer:

The graph of \(f(x)\) has a horizontal asymptote at \(y = 2\) (this statement does not accurately describe the function)