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Let's analyze the given functions [tex]\( f(x) = \ln x \)[/tex] and [tex]\( g(x) = -5 \ln x \)[/tex] to compare their graphs and identify the correct statements from the given options.
1. Domain of the functions:
- The natural logarithm function [tex]\(\ln x\)[/tex] is defined for [tex]\( x > 0 \)[/tex]. Therefore, both functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are defined for [tex]\( x > 0 \)[/tex], i.e., their domain is [tex]\((0, \infty)\)[/tex].
- The domain statement [tex]\(\{x \mid -5 < x < \infty\}\)[/tex] is incorrect because [tex]\(\ln x\)[/tex] and [tex]\(-5 \ln x\)[/tex] are not defined for [tex]\( x \leq 0 \)[/tex].
2. Transformation of the graph of [tex]\( f(x) \)[/tex] to obtain [tex]\( g(x) \)[/tex]:
- The function [tex]\( g(x) = -5 \ln x \)[/tex] can be seen as a transformation of [tex]\( f(x) = \ln x \)[/tex].
- Multiplying [tex]\( \ln x \)[/tex] by [tex]\(-1\)[/tex] reflects the graph over the [tex]\( x \)[/tex]-axis.
- Multiplying the result by 5 stretches the graph vertically by a factor of 5.
- Therefore, the graph of function [tex]\( g \)[/tex] is the graph of function [tex]\( f \)[/tex] reflected over the [tex]\( x \)[/tex]-axis and vertically stretched by a factor of 5. This statement is correct.
3. Behavior of the functions as [tex]\( x \)[/tex] increases:
- For [tex]\( f(x) = \ln x \)[/tex], the function increases as [tex]\( x \)[/tex] increases.
- For [tex]\( g(x) = -5 \ln x \)[/tex], since [tex]\( \ln x \)[/tex] increases, multiplying it by -5 will make [tex]\( g(x) \)[/tex] decrease as [tex]\( x \)[/tex] increases.
- Therefore, the statement that the graph of [tex]\( g \)[/tex] decreases as [tex]\( x \)[/tex] increases, unlike [tex]\( f \)[/tex], is correct.
4. Vertical asymptotes:
- Both [tex]\( \ln x \)[/tex] and [tex]\(-5 \ln x \)[/tex] approach [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] approaches 0 from the right.
- Hence, both functions have a vertical asymptote at [tex]\( x = 0 \)[/tex]. This statement is correct.
5. Y-intercept:
- The natural logarithm function [tex]\( \ln x \)[/tex] is not defined at [tex]\( x = 0 \)[/tex] and neither is [tex]\(-5 \ln x\)[/tex].
- Therefore, neither [tex]\( f \)[/tex] nor [tex]\( g \)[/tex] has a [tex]\( y \)[/tex]-intercept. This statement is incorrect.
Based on the above analysis, the correct answers are:
- The graph of function [tex]\( g \)[/tex] is the graph of function [tex]\( f \)[/tex] reflected over the [tex]\( x \)[/tex]-axis and vertically stretched by a factor of 5.
- Unlike the graph of function [tex]\( f \)[/tex], the graph of function [tex]\( g \)[/tex] decreases as [tex]\( x \)[/tex] increases.
- The graphs of both functions have a vertical asymptote of [tex]\( x = 0 \)[/tex].
Thus, the correct answers are:
[tex]\[ 2. \quad \text{The graph of function } g \text{ is the graph of function } f \text{ reflected over the } x\text{-axis and vertically stretched by a factor of 5.} \][/tex]
[tex]\[ 3. \quad \text{Unlike the graph of function } f, \text{ the graph of function } g \text{ decreases as } x \text{ increases.} \][/tex]
[tex]\[ 4. \quad \text{The graphs of both functions have a vertical asymptote of } x = 0. \][/tex]
Therefore, the correct selections are:
[tex]\[ (2, 3, 4) \][/tex]
1. Domain of the functions:
- The natural logarithm function [tex]\(\ln x\)[/tex] is defined for [tex]\( x > 0 \)[/tex]. Therefore, both functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are defined for [tex]\( x > 0 \)[/tex], i.e., their domain is [tex]\((0, \infty)\)[/tex].
- The domain statement [tex]\(\{x \mid -5 < x < \infty\}\)[/tex] is incorrect because [tex]\(\ln x\)[/tex] and [tex]\(-5 \ln x\)[/tex] are not defined for [tex]\( x \leq 0 \)[/tex].
2. Transformation of the graph of [tex]\( f(x) \)[/tex] to obtain [tex]\( g(x) \)[/tex]:
- The function [tex]\( g(x) = -5 \ln x \)[/tex] can be seen as a transformation of [tex]\( f(x) = \ln x \)[/tex].
- Multiplying [tex]\( \ln x \)[/tex] by [tex]\(-1\)[/tex] reflects the graph over the [tex]\( x \)[/tex]-axis.
- Multiplying the result by 5 stretches the graph vertically by a factor of 5.
- Therefore, the graph of function [tex]\( g \)[/tex] is the graph of function [tex]\( f \)[/tex] reflected over the [tex]\( x \)[/tex]-axis and vertically stretched by a factor of 5. This statement is correct.
3. Behavior of the functions as [tex]\( x \)[/tex] increases:
- For [tex]\( f(x) = \ln x \)[/tex], the function increases as [tex]\( x \)[/tex] increases.
- For [tex]\( g(x) = -5 \ln x \)[/tex], since [tex]\( \ln x \)[/tex] increases, multiplying it by -5 will make [tex]\( g(x) \)[/tex] decrease as [tex]\( x \)[/tex] increases.
- Therefore, the statement that the graph of [tex]\( g \)[/tex] decreases as [tex]\( x \)[/tex] increases, unlike [tex]\( f \)[/tex], is correct.
4. Vertical asymptotes:
- Both [tex]\( \ln x \)[/tex] and [tex]\(-5 \ln x \)[/tex] approach [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] approaches 0 from the right.
- Hence, both functions have a vertical asymptote at [tex]\( x = 0 \)[/tex]. This statement is correct.
5. Y-intercept:
- The natural logarithm function [tex]\( \ln x \)[/tex] is not defined at [tex]\( x = 0 \)[/tex] and neither is [tex]\(-5 \ln x\)[/tex].
- Therefore, neither [tex]\( f \)[/tex] nor [tex]\( g \)[/tex] has a [tex]\( y \)[/tex]-intercept. This statement is incorrect.
Based on the above analysis, the correct answers are:
- The graph of function [tex]\( g \)[/tex] is the graph of function [tex]\( f \)[/tex] reflected over the [tex]\( x \)[/tex]-axis and vertically stretched by a factor of 5.
- Unlike the graph of function [tex]\( f \)[/tex], the graph of function [tex]\( g \)[/tex] decreases as [tex]\( x \)[/tex] increases.
- The graphs of both functions have a vertical asymptote of [tex]\( x = 0 \)[/tex].
Thus, the correct answers are:
[tex]\[ 2. \quad \text{The graph of function } g \text{ is the graph of function } f \text{ reflected over the } x\text{-axis and vertically stretched by a factor of 5.} \][/tex]
[tex]\[ 3. \quad \text{Unlike the graph of function } f, \text{ the graph of function } g \text{ decreases as } x \text{ increases.} \][/tex]
[tex]\[ 4. \quad \text{The graphs of both functions have a vertical asymptote of } x = 0. \][/tex]
Therefore, the correct selections are:
[tex]\[ (2, 3, 4) \][/tex]
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