Question

select the correct answer.
which statement about the end behavior of the logarithmic function ( f(x)=log (x + 3)-2 ) is true?
a. as ( x ) decreases, ( y ) approaches the vertical asymptote at ( x=-3 ).
b. as ( x ) decreases, ( y ) approaches the vertical asymptote at ( x=-1 ).
c. as ( x ) increases, ( y ) approaches negative infinity.
d. as ( x ) decreases, ( y ) approaches positive infinity.

Explanation:

Step1: Find the vertical asymptote

For a logarithmic function \(y = \log(u)\), the vertical asymptote occurs when \(u=0\).
Set \(x + 3=0\), then \(x=-3\).

Step2: Analyze the end - behavior

As \(x\) approaches \(-3\) from the right (\(x\gt - 3\)), \(y=\log(x + 3)-2\) approaches \(-\infty\).
As \(x\) increases (\(x\to+\infty\)), \(y=\log(x + 3)-2\to+\infty\).

Answer:

A. As \(x\) decreases, \(y\) approaches the vertical asymptote at \(x =-3\)