Answer :
To analyze the end behavior of the logarithmic function [tex]\( f(x) = \log(x + 3) - 2 \)[/tex], let's carefully understand its characteristics and behavior at extreme values of [tex]\( x \)[/tex].
1. Vertical Asymptote: The logarithmic function [tex]\( \log(x + 3) \)[/tex] has a vertical asymptote where its argument equals zero. Therefore, we set:
[tex]\[ x + 3 = 0 \implies x = -3 \][/tex]
This means there is a vertical asymptote at [tex]\( x = -3 \)[/tex].
2. End Behavior as [tex]\( x \)[/tex] Decreases:
- When [tex]\( x \)[/tex] approaches the vertical asymptote [tex]\( x = -3 \)[/tex] from the right (i.e., as [tex]\( x \)[/tex] gets closer to -3 but remains greater than -3), the term [tex]\( x + 3 \)[/tex] approaches 0 from the positive side.
- The logarithmic function [tex]\( \log(x + 3) \)[/tex] tends towards negative infinity as [tex]\( x + 3 \)[/tex] approaches 0 from the positive side.
- Consequently, [tex]\( \log(x + 3) - 2 \)[/tex] will also approach negative infinity.
Therefore, as [tex]\( x \)[/tex] decreases and approaches -3 from the right, the value of [tex]\( y \)[/tex] (or [tex]\( f(x) \)[/tex]) approaches negative infinity.
Given the above analysis, let’s compare this with the provided choices:
A. As [tex]\( x \)[/tex] decreases, [tex]\( y \)[/tex] approaches the vertical asymptote at [tex]\( x = -3 \)[/tex].
- This statement correctly describes the behavior near the vertical asymptote.
B. As [tex]\( x \)[/tex] decreases, [tex]\( y \)[/tex] approaches the vertical asymptote at [tex]\( x = -1 \)[/tex].
- This is incorrect because the vertical asymptote is at [tex]\( x = -3 \)[/tex], not [tex]\( x = -1 \)[/tex].
C. As [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] approaches negative infinity.
- This is incorrect since as [tex]\( x \)[/tex] increases, [tex]\( \log(x + 3) - 2 \)[/tex] actually increases (because logarithmic functions increase as their argument increases).
D. As [tex]\( x \)[/tex] decreases, [tex]\( y \)[/tex] approaches positive infinity.
- This is incorrect because as [tex]\( x \)[/tex] decreases towards the vertical asymptote [tex]\( x = -3 \)[/tex], [tex]\( y \)[/tex] tends towards negative infinity, not positive infinity.
Thus, the correct answer is:
A. As [tex]\( x \)[/tex] decreases, [tex]\( y \)[/tex] approaches the vertical asymptote at [tex]\( x = -3 \)[/tex].
1. Vertical Asymptote: The logarithmic function [tex]\( \log(x + 3) \)[/tex] has a vertical asymptote where its argument equals zero. Therefore, we set:
[tex]\[ x + 3 = 0 \implies x = -3 \][/tex]
This means there is a vertical asymptote at [tex]\( x = -3 \)[/tex].
2. End Behavior as [tex]\( x \)[/tex] Decreases:
- When [tex]\( x \)[/tex] approaches the vertical asymptote [tex]\( x = -3 \)[/tex] from the right (i.e., as [tex]\( x \)[/tex] gets closer to -3 but remains greater than -3), the term [tex]\( x + 3 \)[/tex] approaches 0 from the positive side.
- The logarithmic function [tex]\( \log(x + 3) \)[/tex] tends towards negative infinity as [tex]\( x + 3 \)[/tex] approaches 0 from the positive side.
- Consequently, [tex]\( \log(x + 3) - 2 \)[/tex] will also approach negative infinity.
Therefore, as [tex]\( x \)[/tex] decreases and approaches -3 from the right, the value of [tex]\( y \)[/tex] (or [tex]\( f(x) \)[/tex]) approaches negative infinity.
Given the above analysis, let’s compare this with the provided choices:
A. As [tex]\( x \)[/tex] decreases, [tex]\( y \)[/tex] approaches the vertical asymptote at [tex]\( x = -3 \)[/tex].
- This statement correctly describes the behavior near the vertical asymptote.
B. As [tex]\( x \)[/tex] decreases, [tex]\( y \)[/tex] approaches the vertical asymptote at [tex]\( x = -1 \)[/tex].
- This is incorrect because the vertical asymptote is at [tex]\( x = -3 \)[/tex], not [tex]\( x = -1 \)[/tex].
C. As [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] approaches negative infinity.
- This is incorrect since as [tex]\( x \)[/tex] increases, [tex]\( \log(x + 3) - 2 \)[/tex] actually increases (because logarithmic functions increase as their argument increases).
D. As [tex]\( x \)[/tex] decreases, [tex]\( y \)[/tex] approaches positive infinity.
- This is incorrect because as [tex]\( x \)[/tex] decreases towards the vertical asymptote [tex]\( x = -3 \)[/tex], [tex]\( y \)[/tex] tends towards negative infinity, not positive infinity.
Thus, the correct answer is:
A. As [tex]\( x \)[/tex] decreases, [tex]\( y \)[/tex] approaches the vertical asymptote at [tex]\( x = -3 \)[/tex].
