To analyze the end behavior of the logarithmic function [tex]\( f(x) = \log(x + 3) - 2 \)[/tex], we need to understand the properties and behavior of logarithmic functions in general.
Step-by-step:
1. Identify the argument of the logarithmic function:
The given function is [tex]\( f(x) = \log(x + 3) - 2 \)[/tex]. Within this expression, the logarithmic part is [tex]\( \log(x + 3) \)[/tex].
2. Determine the domain of the logarithmic function:
A logarithmic function [tex]\( \log(a) \)[/tex] is defined only for [tex]\( a > 0 \)[/tex]. In this case, the argument [tex]\( x + 3 \)[/tex] must be greater than 0. Therefore:
[tex]\[
x + 3 > 0 \implies x > -3
\][/tex]
This tells us that the function [tex]\( f(x) \)[/tex] is only defined for [tex]\( x > -3 \)[/tex].
3. Identify the vertical asymptote:
The vertical asymptote of a logarithmic function occurs where its argument approaches zero. Here, the argument is [tex]\( x + 3 \)[/tex], so the vertical asymptote occurs where:
[tex]\[
x + 3 = 0 \implies x = -3
\][/tex]
4. Analyze the behavior near the vertical asymptote:
As [tex]\( x \)[/tex] approaches [tex]\(-3\)[/tex] from the right (i.e., [tex]\( x \to -3^+ \)[/tex]), the argument of the logarithmic function [tex]\( x + 3 \)[/tex] approaches zero from the positive side. We know from the properties of logarithmic functions that as their argument approaches zero from the positive side, the logarithm tends to negative infinity:
[tex]\[
\log(x + 3) \to -\infty \text{ as } x \to -3^+
\][/tex]
Consequently, [tex]\( f(x) = \log(x + 3) - 2 \)[/tex] will also tend to negative infinity as [tex]\( x \)[/tex] approaches [tex]\(-3\)[/tex]:
[tex]\[
f(x) = \log(x + 3) - 2 \to -\infty - 2 = -\infty \text{ as } x \to -3^+
\][/tex]
In summary, the correct statement about the end behavior of the 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.
