Answered

At Westonci.ca, we connect you with the answers you need, thanks to our active and informed community. Get quick and reliable answers to your questions from a dedicated community of professionals on our platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.

Select the correct answer.

Which statement about the end behavior of the logarithmic function [tex]f(x)=\log (x+3)-2[/tex] is true?

A. As [tex]x[/tex] decreases to the vertical asymptote at [tex]x=-3[/tex], [tex]y[/tex] decreases to negative infinity.
B. As [tex]x[/tex] decreases to the vertical asymptote at [tex]x=-1[/tex], [tex]y[/tex] decreases to negative infinity.
C. As [tex]x[/tex] decreases to the vertical asymptote at [tex]x=-3[/tex], [tex]y[/tex] increases to positive infinity.
D. As [tex]x[/tex] decreases to the vertical asymptote at [tex]x=-1[/tex], [tex]y[/tex] increases to positive infinity.

Sagot :

GinnyAnswer
To determine the correct statement about the end behavior of the logarithmic function [tex]\( f(x) = \log(x+3) - 2 \)[/tex], let's break down the problem step by step.

### Step 1: Identify the Vertical Asymptote
The general form of a logarithmic function is [tex]\( \log_b(x - c) + d \)[/tex], where the vertical asymptote occurs at [tex]\( x = c \)[/tex]. For the given function [tex]\( f(x) = \log(x + 3) - 2 \)[/tex]:

[tex]\[ \log(x + 3) \][/tex]
The logarithmic argument [tex]\( x + 3 \)[/tex] must be greater than zero (since the logarithm of a non-positive number is undefined). Therefore, we set up the inequality:

[tex]\[ x + 3 > 0 \][/tex]
[tex]\[ x > -3 \][/tex]

So, the vertical asymptote is at [tex]\( x = -3 \)[/tex], since as [tex]\( x \)[/tex] approaches [tex]\( -3 \)[/tex] from the right, [tex]\( x + 3 \)[/tex] approaches zero.

### Step 2: Analyze the Behavior Near the Vertical Asymptote
Next, consider the behavior of the function as [tex]\( x \)[/tex] approaches the vertical asymptote [tex]\( -3 \)[/tex] from the right.

For values of [tex]\( x \)[/tex] approaching [tex]\( -3 \)[/tex] from the right (i.e., [tex]\( x > -3 \)[/tex]):
[tex]\[ \log(x + 3) \][/tex]

Since [tex]\( x + 3 \)[/tex] is a small positive number approaching zero, [tex]\( \log(x + 3) \)[/tex] approaches [tex]\( - \infty \)[/tex]. Therefore, as [tex]\( x \)[/tex] nears [tex]\( -3 \)[/tex]:

[tex]\[ \log(x + 3) \rightarrow -\infty \][/tex]

Using this in our function:
[tex]\[ f(x) = \log(x + 3) - 2 \][/tex]

[tex]\[ f(x) \rightarrow -\infty - 2 \][/tex]

[tex]\[ f(x) \rightarrow -\infty \][/tex]

Thus, as [tex]\( x \)[/tex] approaches [tex]\( -3 \)[/tex] from the right, [tex]\( f(x) \)[/tex] decreases towards [tex]\( -\infty \)[/tex].

### Conclusion
Given these observations, the correct statement about the end behavior of the logarithmic function [tex]\( f(x) = \log(x + 3) - 2 \)[/tex] is:

A. As [tex]\( x \)[/tex] decreases to the vertical asymptote at [tex]\( x = -3 \)[/tex], [tex]\( y \)[/tex] decreases to negative infinity.

So, the correct answer is:
[tex]\[ \boxed{A} \][/tex]
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.