Question

select the correct answer.
which function has an asymptote at ( x = 5 ) and an ( x )-intercept of ( (6,0) )?

○ a. ( f(x)=log (x - 5) )
○ b. ( f(x)=log (x + 5) )
○ c. ( f(x)=log x - 5 )
○ d. ( f(x)=log x + 5 )

Explanation:

Step1: Analyze the vertical asymptote

For the function \(y = \log(u)\), the vertical asymptote is at \(u = 0\).
For option A: \(u=x - 5\), the vertical asymptote is \(x-5=0\), so \(x = 5\).
For option B: \(u=x + 5\), the vertical asymptote is \(x+5=0\), so \(x=-5\).
For option C: \(y=\log x-5\), the vertical asymptote is \(x = 0\).
For option D: \(y=\log x + 5\), the vertical asymptote is \(x = 0\).

Step2: Analyze the \(x\) - intercept

The \(x\) - intercept occurs when \(y = 0\).
For option A: Set \(y=\log(x - 5)=0\). Since \(\log a=0\) implies \(a = 1\) (because \(\log_{10}1=0\)), then \(x-5=1\), so \(x=6\).

Answer:

A. \(f(x)=\log(x - 5)\)