Find the best solutions to your questions at Westonci.ca, the premier Q&A platform with a community of knowledgeable experts. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
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]
### 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]
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.