To determine the end behavior of the logarithmic function [tex]\( f(x) = \log(x + 3) - 2 \)[/tex], we need to understand how the function behaves as [tex]\( x \)[/tex] approaches certain critical values.
1. Identify the Vertical Asymptote:
A logarithmic function has a vertical asymptote at the value where the argument inside the logarithm becomes zero. The argument inside the logarithm is [tex]\( x + 3 \)[/tex].
Therefore, set the argument to zero:
[tex]\[
x + 3 = 0 \implies x = -3
\][/tex]
The function [tex]\( f(x) = \log(x + 3) - 2 \)[/tex] has a vertical asymptote at [tex]\( x = -3 \)[/tex].
2. Determine the Behavior Near the Vertical Asymptote:
Examine how [tex]\( f(x) \)[/tex] behaves as [tex]\( x \)[/tex] approaches [tex]\( -3 \)[/tex] from the right ( [tex]\( x \to -3^+ \)[/tex] ).
- When [tex]\( x \)[/tex] gets very close to [tex]\( -3 \)[/tex] from the right side, [tex]\( x + 3 \)[/tex] becomes a very small positive number.
- The logarithm of a very small positive number approaches negative infinity.
Thus:
[tex]\[
\log(x + 3) \to -\infty \text{ as } x \to -3^+
\][/tex]
Given that [tex]\( f(x) = \log(x + 3) - 2 \)[/tex], the subtraction of 2 doesn’t change the fact that [tex]\( f(x) \)[/tex] will approach negative infinity under this condition.
Therefore:
[tex]\[
f(x) = \log(x + 3) - 2 \to -\infty \text{ as } x \to -3^+
\][/tex]
3. Select the Correct Statement:
Let's match the above analysis with the provided answer choices:
- A: As [tex]\( x \)[/tex] decreases to the vertical asymptote at [tex]\( x = -3 \)[/tex], [tex]\( y \)[/tex] decreases to negative infinity. (This is correct)
- B: Incorrect, since [tex]\( x = -1 \)[/tex] is not the vertical asymptote.
- C: Incorrect, [tex]\( y \)[/tex] decreases to negative infinity, not increases to positive infinity.
- D: Incorrect for reasons similar to B and C.
Therefore, the correct statement is:
A. As [tex]\( x \)[/tex] decreases to the vertical asymptote at [tex]\( x = -3 \)[/tex], [tex]\( y \)[/tex] decreases to negative infinity.
