Question

question 6
which of the following is the asymptote of the function ( f(x)=log _{2}(x + 3)-1 )?
a ( x=-1 )
b ( y=-3 )
c ( y=-1 )
d ( x=-3 )
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Explanation:

Step1: Recall log function asymptote

For a logarithmic function \( f(x)=\log_b(x - h)+k \), the vertical asymptote is \( x = h \) (since the argument \( x - h>0 \), so \( x = h \) is the vertical asymptote).
In the function \( f(x)=\log_2(x + 3)-1 \), we can rewrite the argument as \( x-(-3) \), so comparing with \( \log_b(x - h)+k \), we have \( h=-3 \).

Step2: Determine the asymptote

So the vertical asymptote of the function \( f(x)=\log_2(x + 3)-1 \) is \( x=-3 \).
Now let's analyze the options:

• Option A: \( x = - 1 \) is not the vertical asymptote.
• Option B: \( y=-3 \) is not an asymptote (logarithmic functions have vertical asymptotes, not horizontal asymptotes in the way this is presented, and the horizontal shift/vertical shift doesn't give \( y = - 3 \)).
• Option C: \( y=-1 \) is the vertical shift, not an asymptote (logarithmic functions have vertical asymptotes, and the horizontal asymptote for logarithmic functions doesn't exist in the same way as for exponential functions; the \( - 1 \) is a vertical shift of the graph).
• Option D: \( x=-3 \) matches the vertical asymptote we found.

Answer:

D. \( x = - 3 \)