To analyze the behavior and transformation of the functions [tex]\( f(x) = \ln x \)[/tex] and [tex]\( g(x) = -5 \ln x \)[/tex], let's break down the properties and compare the two functions in various aspects:
1. Vertical Asymptote:
- For the function [tex]\( f(x) = \ln x \)[/tex], there is a vertical asymptote at [tex]\( x = 0 \)[/tex]. This is because the natural logarithm function tends to [tex]\(-\infty\)[/tex] as [tex]\( x \)[/tex] approaches 0 from the positive side.
- For the function [tex]\( g(x) = -5 \ln x \)[/tex], the vertical asymptote remains at [tex]\( x = 0 \)[/tex]. This is because the transformation applied to [tex]\( f(x) \)[/tex] does not affect the vertical asymptote.
Therefore, the graph of both functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] have a vertical asymptote at [tex]\( x = 0 \)[/tex].
2. Monotonicity:
- The function [tex]\( f(x) = \ln x \)[/tex] is an increasing function, meaning as [tex]\( x \)[/tex] increases, [tex]\( \ln x \)[/tex] also increases.
- The function [tex]\( g(x) = -5 \ln x \)[/tex] is a decreasing function. This is because multiplying the natural logarithm by [tex]\(-5\)[/tex] reverses the direction of the slope, causing it to decrease as [tex]\( x \)[/tex] increases.
So, unlike the graph of function [tex]\( f \)[/tex], the graph of [tex]\( g \)[/tex] decreases as [tex]\( x \)[/tex] increases.
3. Transformation:
- The function [tex]\( g(x) = -5 \ln x \)[/tex] can be considered as a transformation of [tex]\( f(x) = \ln x \)[/tex]. Specifically, [tex]\( g(x) \)[/tex] is obtained by reflecting [tex]\( f(x) \)[/tex] over the [tex]\( x \)[/tex]-axis and then stretching it vertically by a factor of 5.
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.
4. Domain:
- Both functions [tex]\( f(x) = \ln x \)[/tex] and [tex]\( g(x) = -5 \ln x \)[/tex] have the same domain: [tex]\( x > 0 \)[/tex]. Therefore, the statements claiming different domains for [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are incorrect.
The correct domain for both functions is [tex]\( \{ x \mid x > 0 \} \)[/tex].
Summary:
- The vertical asymptote for both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] is [tex]\( x = 0 \)[/tex].
- [tex]\( f(x) \)[/tex] is an increasing function, whereas [tex]\( g(x) \)[/tex] is a decreasing function.
- The graph of [tex]\( g(x) \)[/tex] is the reflection of [tex]\( f(x) \)[/tex] over the [tex]\( x \)[/tex]-axis and a vertical stretch by a factor of 5.
Thus, the correct descriptive statements for comparing [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are:
1. The graphs of both functions have a vertical asymptote of [tex]\( x = 0 \)[/tex].
2. Unlike the graph of function [tex]\( f \)[/tex], the graph of function [tex]\( g \)[/tex] decreases as [tex]\( x \)[/tex] increases.
3. 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.
