Answer :
To compare the graphs of the functions [tex]\( f(x) = \ln x \)[/tex] and [tex]\( g(x) = -5 \ln x \)[/tex], we will examine their properties such as vertical asymptotes, behavior as [tex]\( x \)[/tex] increases, and transformations.
1. Vertical Asymptote:
- The function [tex]\( \ln x \)[/tex] has a vertical asymptote at [tex]\( x = 0 \)[/tex]. This is because the natural logarithm function is undefined for [tex]\( x \leq 0 \)[/tex] and approaches [tex]\(-\infty\)[/tex] as [tex]\( x \)[/tex] approaches 0 from the positive side.
- Since [tex]\( g(x) = -5 \ln x \)[/tex] is also a logarithmic function (scaled and reflected), it will share the same vertical asymptote at [tex]\( x = 0 \)[/tex].
- Therefore, "The graphs of both functions have a vertical asymptote of [tex]\( x=0 \)[/tex]" is correct.
2. Behavior as [tex]\( x \)[/tex] Increases:
- The function [tex]\( f(x) = \ln x \)[/tex] is an increasing function. As [tex]\( x \)[/tex] increases, [tex]\( \ln x \)[/tex] increases.
- However, [tex]\( g(x) = -5 \ln x \)[/tex] has a negative coefficient in front of [tex]\( \ln x \)[/tex], which causes [tex]\( g(x) \)[/tex] to decrease as [tex]\( x \)[/tex] increases. The negative coefficient reflects the graph over the [tex]\( x \)[/tex]-axis, and the factor of 5 vertically stretches it.
- Therefore, "Unlike the graph of function [tex]\( f \)[/tex], the graph of function [tex]\( g \)[/tex] decreases as [tex]\( x \)[/tex] increases" is correct.
3. Y-intercept:
- The function [tex]\( \ln x \)[/tex] does not have a [tex]\( y \)[/tex]-intercept because it is undefined at [tex]\( x = 0 \)[/tex].
- Similarly, [tex]\( g(x) = -5 \ln x \)[/tex] will also not have a [tex]\( y \)[/tex]-intercept for the same reason.
- Therefore, "Unlike the graph of function [tex]\( f \)[/tex], the graph of function [tex]\( g \)[/tex] has a [tex]\( y \)[/tex]-intercept" is incorrect.
4. Domain:
- The domain of [tex]\( f(x) \)[/tex] is [tex]\( (0, \infty) \)[/tex], meaning it is defined for all positive real numbers.
- The domain of [tex]\( g(x) = -5 \ln x \)[/tex] is also [tex]\( (0, \infty) \)[/tex] because scaling and reflecting the log function does not change the set of [tex]\( x \)[/tex]-values for which it is defined.
- Therefore, "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]" is incorrect.
5. Graph Transformation:
- The function [tex]\( g(x) = -5 \ln x \)[/tex] can be seen as a transformation of [tex]\( f(x) = \ln x \)[/tex]. Specifically:
- The negative sign indicates a reflection over the [tex]\( x \)[/tex]-axis.
- The factor of 5 indicates a vertical stretch 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" is correct.
Summarizing these points, the correct answers are:
- 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].
- 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.
1. Vertical Asymptote:
- The function [tex]\( \ln x \)[/tex] has a vertical asymptote at [tex]\( x = 0 \)[/tex]. This is because the natural logarithm function is undefined for [tex]\( x \leq 0 \)[/tex] and approaches [tex]\(-\infty\)[/tex] as [tex]\( x \)[/tex] approaches 0 from the positive side.
- Since [tex]\( g(x) = -5 \ln x \)[/tex] is also a logarithmic function (scaled and reflected), it will share the same vertical asymptote at [tex]\( x = 0 \)[/tex].
- Therefore, "The graphs of both functions have a vertical asymptote of [tex]\( x=0 \)[/tex]" is correct.
2. Behavior as [tex]\( x \)[/tex] Increases:
- The function [tex]\( f(x) = \ln x \)[/tex] is an increasing function. As [tex]\( x \)[/tex] increases, [tex]\( \ln x \)[/tex] increases.
- However, [tex]\( g(x) = -5 \ln x \)[/tex] has a negative coefficient in front of [tex]\( \ln x \)[/tex], which causes [tex]\( g(x) \)[/tex] to decrease as [tex]\( x \)[/tex] increases. The negative coefficient reflects the graph over the [tex]\( x \)[/tex]-axis, and the factor of 5 vertically stretches it.
- Therefore, "Unlike the graph of function [tex]\( f \)[/tex], the graph of function [tex]\( g \)[/tex] decreases as [tex]\( x \)[/tex] increases" is correct.
3. Y-intercept:
- The function [tex]\( \ln x \)[/tex] does not have a [tex]\( y \)[/tex]-intercept because it is undefined at [tex]\( x = 0 \)[/tex].
- Similarly, [tex]\( g(x) = -5 \ln x \)[/tex] will also not have a [tex]\( y \)[/tex]-intercept for the same reason.
- Therefore, "Unlike the graph of function [tex]\( f \)[/tex], the graph of function [tex]\( g \)[/tex] has a [tex]\( y \)[/tex]-intercept" is incorrect.
4. Domain:
- The domain of [tex]\( f(x) \)[/tex] is [tex]\( (0, \infty) \)[/tex], meaning it is defined for all positive real numbers.
- The domain of [tex]\( g(x) = -5 \ln x \)[/tex] is also [tex]\( (0, \infty) \)[/tex] because scaling and reflecting the log function does not change the set of [tex]\( x \)[/tex]-values for which it is defined.
- Therefore, "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]" is incorrect.
5. Graph Transformation:
- The function [tex]\( g(x) = -5 \ln x \)[/tex] can be seen as a transformation of [tex]\( f(x) = \ln x \)[/tex]. Specifically:
- The negative sign indicates a reflection over the [tex]\( x \)[/tex]-axis.
- The factor of 5 indicates a vertical stretch 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" is correct.
Summarizing these points, the correct answers are:
- 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].
- 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.
