To determine the correct answers, let's evaluate how the function [tex]\( g(x) = -5 \ln x \)[/tex] compares with the function [tex]\( f(x) = \ln x \)[/tex].
1. 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 true. Reflecting [tex]\( f(x) \)[/tex] over the [tex]\( x \)[/tex]-axis gives us [tex]\( -\ln x \)[/tex], and multiplying it by 5 stretches it vertically by a factor of 5, resulting in [tex]\( g(x) = -5 \ln x \)[/tex].
2. Unlike the graph of function [tex]\( f \)[/tex], the graph of function [tex]\( g \)[/tex] decreases as [tex]\( x \)[/tex] increases.
This statement is also true. The function [tex]\( f(x) = \ln x \)[/tex] increases as [tex]\( x \)[/tex] increases, while [tex]\( g(x) = -5 \ln x \)[/tex] decreases as [tex]\( x \)[/tex] increases because of the negative coefficient.
3. The graphs of both functions have a vertical asymptote of [tex]\( x = 0 \)[/tex].
This statement is true. Both [tex]\( \ln x \)[/tex] and [tex]\( -5 \ln x \)[/tex] have a vertical asymptote at [tex]\( x = 0 \)[/tex].
4. Unlike the graph of function [tex]\( f \)[/tex], the graph of function [tex]\( g \)[/tex] has a domain of [tex]\( \{x \mid -5 < x < \infty\} \)[/tex].
This statement is false. The domain of [tex]\( f(x) = \ln x \)[/tex] is [tex]\( x > 0 \)[/tex]. Similarly, the domain of [tex]\( g(x) = -5 \ln x \)[/tex] is also [tex]\( x > 0 \)[/tex]. The domain given [tex]\( \{x \mid -5 < x < \infty\} \)[/tex] is incorrect and does not apply to [tex]\( g(x) \)[/tex].
5. Unlike the graph of function [tex]\( f \)[/tex], the graph of function [tex]\( g \)[/tex] has a [tex]\( y \)[/tex]-intercept.
This statement is false. Neither [tex]\( f(x) = \ln x \)[/tex] nor [tex]\( g(x) = -5 \ln x \)[/tex] have a [tex]\( y \)[/tex]-intercept because they are both undefined at [tex]\( x = 0 \)[/tex].
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].
