Answered

Welcome to Westonci.ca, the place where your questions are answered by a community of knowledgeable contributors. Our platform offers a seamless experience for finding reliable answers from a network of experienced professionals. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.

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 hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.