select the correct answer. which statement about the end - behavior of the logarithmic function (f(x)=log(x…

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.

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.

Answer

Explanation:

Step1: Recall the domain of logarithmic functions

The domain of $y = \log(x + 3)-2$ is $x+3>0$, or $x>- 3$. The vertical - asymptote of the function $y=\log(x + 3)-2$ is $x=-3$.

Step2: Analyze the end - behavior as $x$ decreases

As $x$ decreases, it gets closer and closer to the vertical asymptote $x = - 3$. Since the domain of the logarithmic function $y=\log(x + 3)-2$ is $x>-3$, as $x$ approaches $-3$ from the right (decreasing values of $x$ that are still in the domain), $y$ approaches negative infinity. As $x$ decreases towards the vertical asymptote $x=-3$, the function value $y$ approaches the vertical asymptote at $x = - 3$.

Step3: Analyze the end - behavior as $x$ increases

As $x$ increases, $\log(x + 3)$ also increases. So, as $x$ increases, $y=\log(x + 3)-2$ approaches positive infinity.

Answer:

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