IDNLearn.com offers a unique blend of expert answers and community-driven knowledge. Ask any question and receive comprehensive, well-informed responses from our dedicated team of experts.
Sagot :
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.
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.
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Find precise solutions at IDNLearn.com. Thank you for trusting us with your queries, and we hope to see you again.
