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To analyze the end behavior of the logarithmic function [tex]\( f(x) = \log(x + 3) - 2 \)[/tex], let's break it down in a step-by-step manner:
1. Understanding the Argument of the Logarithm:
- The function [tex]\( f(x) = \log(x + 3) - 2 \)[/tex] contains the logarithmic expression [tex]\( \log(x + 3) \)[/tex]. For this expression to be defined, the argument must be greater than zero:
[tex]\[ x + 3 > 0 \quad \text{or} \quad x > -3 \][/tex]
- Therefore, the vertical asymptote of the function occurs at [tex]\( x = -3 \)[/tex], because as [tex]\( x \)[/tex] approaches this value from the right (i.e., [tex]\( x \to -3^+ \)[/tex]), the argument of the logarithm approaches zero from the positive side, making the logarithm approach negative infinity.
2. End Behavior as [tex]\( x \)[/tex] Decreases:
- When [tex]\( x \)[/tex] approaches [tex]\(-3\)[/tex] from the right (i.e., [tex]\( x \to -3^+ \)[/tex]), the term [tex]\( x + 3 \)[/tex] becomes very small and positive. Hence, [tex]\( \log(x + 3) \)[/tex] becomes very large in the negative direction (i.e., tends to [tex]\(-\infty\)[/tex]).
- Consequently, [tex]\( f(x) = \log(x + 3) - 2 \)[/tex] will also tend to [tex]\(-\infty\)[/tex] as [tex]\( x \to -3^+ \)[/tex], since subtracting 2 from a very large negative number remains a large negative number.
- This behavior demonstrates that [tex]\( y \)[/tex] approaches [tex]\(-\infty\)[/tex] and [tex]\( f(x) \)[/tex] approaches the vertical asymptote at [tex]\( x = -3 \)[/tex] as [tex]\( x \)[/tex] decreases.
3. Incorrect Statements Analysis:
- Option B is incorrect because it wrongly identifies the vertical asymptote as [tex]\( x = -1 \)[/tex]. The correct vertical asymptote is [tex]\( x = -3 \)[/tex].
- Option C is incorrect because it suggests that as [tex]\( x \)[/tex] increases, [tex]\( y \rightarrow -\infty \)[/tex], which is not true. For large values of [tex]\( x \)[/tex], the function will increase slowly since the logarithmic function grows without bound but very slowly.
- Option D is incorrect because it states that as [tex]\( x \)[/tex] decreases, [tex]\( y \rightarrow \infty \)[/tex]. This is not correct since the behavior as [tex]\( x \)[/tex] decreases towards [tex]\(-3\)[/tex] results in [tex]\( y \)[/tex] going to [tex]\(-\infty\)[/tex].
Given these points, the correct statement about the end behavior of the logarithmic function [tex]\( f(x) = \log(x + 3) - 2 \)[/tex] is:
Answer:
[tex]\[ \text{A. As } x \text{ decreases, } y \text{ approaches the vertical asymptote at } x = -3. \][/tex]
1. Understanding the Argument of the Logarithm:
- The function [tex]\( f(x) = \log(x + 3) - 2 \)[/tex] contains the logarithmic expression [tex]\( \log(x + 3) \)[/tex]. For this expression to be defined, the argument must be greater than zero:
[tex]\[ x + 3 > 0 \quad \text{or} \quad x > -3 \][/tex]
- Therefore, the vertical asymptote of the function occurs at [tex]\( x = -3 \)[/tex], because as [tex]\( x \)[/tex] approaches this value from the right (i.e., [tex]\( x \to -3^+ \)[/tex]), the argument of the logarithm approaches zero from the positive side, making the logarithm approach negative infinity.
2. End Behavior as [tex]\( x \)[/tex] Decreases:
- When [tex]\( x \)[/tex] approaches [tex]\(-3\)[/tex] from the right (i.e., [tex]\( x \to -3^+ \)[/tex]), the term [tex]\( x + 3 \)[/tex] becomes very small and positive. Hence, [tex]\( \log(x + 3) \)[/tex] becomes very large in the negative direction (i.e., tends to [tex]\(-\infty\)[/tex]).
- Consequently, [tex]\( f(x) = \log(x + 3) - 2 \)[/tex] will also tend to [tex]\(-\infty\)[/tex] as [tex]\( x \to -3^+ \)[/tex], since subtracting 2 from a very large negative number remains a large negative number.
- This behavior demonstrates that [tex]\( y \)[/tex] approaches [tex]\(-\infty\)[/tex] and [tex]\( f(x) \)[/tex] approaches the vertical asymptote at [tex]\( x = -3 \)[/tex] as [tex]\( x \)[/tex] decreases.
3. Incorrect Statements Analysis:
- Option B is incorrect because it wrongly identifies the vertical asymptote as [tex]\( x = -1 \)[/tex]. The correct vertical asymptote is [tex]\( x = -3 \)[/tex].
- Option C is incorrect because it suggests that as [tex]\( x \)[/tex] increases, [tex]\( y \rightarrow -\infty \)[/tex], which is not true. For large values of [tex]\( x \)[/tex], the function will increase slowly since the logarithmic function grows without bound but very slowly.
- Option D is incorrect because it states that as [tex]\( x \)[/tex] decreases, [tex]\( y \rightarrow \infty \)[/tex]. This is not correct since the behavior as [tex]\( x \)[/tex] decreases towards [tex]\(-3\)[/tex] results in [tex]\( y \)[/tex] going to [tex]\(-\infty\)[/tex].
Given these points, the correct statement about the end behavior of the logarithmic function [tex]\( f(x) = \log(x + 3) - 2 \)[/tex] is:
Answer:
[tex]\[ \text{A. As } x \text{ decreases, } y \text{ approaches the vertical asymptote at } x = -3. \][/tex]
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