Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
To determine which equation could represent the function [tex]\( n \)[/tex] given the conditions, we need to carefully consider how the transformation would affect the rational function [tex]\( m \)[/tex].
### Given Information:
1. The graphs of rational functions [tex]\( m \)[/tex] and [tex]\( n \)[/tex] have the same vertical asymptotes.
2. Both functions have a single [tex]\( x \)[/tex]-intercept at [tex]\( x = 5 \)[/tex].
#### Step-by-Step Analysis:
1. Vertical Asymptotes:
- The vertical asymptotes suggest that the unknown [tex]\( n(x) \)[/tex] should preserve the same factors in the denominator as [tex]\( m(x) \)[/tex] to ensure the asymptotic behavior is the same.
2. [tex]\( x \)[/tex]-Intercept at [tex]\( x = 5 \)[/tex]:
- The [tex]\( x \)[/tex]-intercept for a function [tex]\( f(x) \)[/tex] is found by setting [tex]\( f(x) = 0 \)[/tex]. For [tex]\( n(x) \)[/tex] to have its [tex]\( x \)[/tex]-intercept at [tex]\( x = 5 \)[/tex], we need [tex]\( n(5) = 0 \)[/tex].
#### Evaluating the Options:
Option A: [tex]\( n(x) = 5 m(x) \)[/tex]
- Scaling [tex]\( m(x) \)[/tex] by a factor of 5 will not affect the location of the [tex]\( x \)[/tex]-intercept but it also won't change the vertical asymptotes. However, this does not fix the [tex]\( x \)[/tex]-intercept directly to [tex]\( x = 5 \)[/tex]. It maintains the given [tex]\( x \)[/tex]-intercept of [tex]\( m(x) \)[/tex] but doesn't solve the problem as required.
Option B: [tex]\( n(x) = m(x) + 5 \)[/tex]
- Adding 5 to [tex]\( m(x) \)[/tex] shifts the graph vertically upwards or downwards, which alters the intercepts and affects the graph's shape overall. This would change the [tex]\( x \)[/tex]-intercept location, which does not meet the required condition.
Option C: [tex]\( n(x) = m(x - 5) \)[/tex]
- This option represents a horizontal shift of the function [tex]\( m(x) \)[/tex] to the right by 5 units. To confirm, substituting [tex]\( x = 5 \)[/tex]:
[tex]\[ n(5) = m(5 - 5) = m(0) \][/tex]
For [tex]\( n(x) \)[/tex] to have an [tex]\( x \)[/tex]-intercept at [tex]\( x = 5 \)[/tex]:
[tex]\[ n(5) = 0 \implies m(0) = 0 \][/tex]
Therefore, the intercept at [tex]\( x = 5 \)[/tex] for [tex]\( n \)[/tex] would hold if [tex]\( m \)[/tex] originally had an intercept at [tex]\( x = 0 \)[/tex]. This transformation ensures the [tex]\( x \)[/tex]-intercept is set at [tex]\( x = 5 \)[/tex] appropriately.
Option D: [tex]\( n(x) = m(x + 5) \)[/tex]
- This option would move the function [tex]\( m(x) \)[/tex] left by 5 units. Substituting [tex]\( x = 5 \)[/tex]:
[tex]\[ n(5) = m(5 + 5) = m(10) \][/tex]
Which means that for [tex]\( n(x) \)[/tex] to be zero at [tex]\( x = 5 \)[/tex], [tex]\( m(10) \)[/tex] would need to be zero, which does not ensure that [tex]\( x = 5 \)[/tex] is the intercept in the context provided.
Considering these options, the correct answer is:
[tex]\[ \boxed{C} \][/tex]
This option correctly adjusts the input of [tex]\( m(x) \)[/tex] in a manner that ensures both the vertical asymptotes remain the same and the [tex]\( x \)[/tex]-intercept is specifically located at [tex]\( x = 5 \)[/tex].
### Given Information:
1. The graphs of rational functions [tex]\( m \)[/tex] and [tex]\( n \)[/tex] have the same vertical asymptotes.
2. Both functions have a single [tex]\( x \)[/tex]-intercept at [tex]\( x = 5 \)[/tex].
#### Step-by-Step Analysis:
1. Vertical Asymptotes:
- The vertical asymptotes suggest that the unknown [tex]\( n(x) \)[/tex] should preserve the same factors in the denominator as [tex]\( m(x) \)[/tex] to ensure the asymptotic behavior is the same.
2. [tex]\( x \)[/tex]-Intercept at [tex]\( x = 5 \)[/tex]:
- The [tex]\( x \)[/tex]-intercept for a function [tex]\( f(x) \)[/tex] is found by setting [tex]\( f(x) = 0 \)[/tex]. For [tex]\( n(x) \)[/tex] to have its [tex]\( x \)[/tex]-intercept at [tex]\( x = 5 \)[/tex], we need [tex]\( n(5) = 0 \)[/tex].
#### Evaluating the Options:
Option A: [tex]\( n(x) = 5 m(x) \)[/tex]
- Scaling [tex]\( m(x) \)[/tex] by a factor of 5 will not affect the location of the [tex]\( x \)[/tex]-intercept but it also won't change the vertical asymptotes. However, this does not fix the [tex]\( x \)[/tex]-intercept directly to [tex]\( x = 5 \)[/tex]. It maintains the given [tex]\( x \)[/tex]-intercept of [tex]\( m(x) \)[/tex] but doesn't solve the problem as required.
Option B: [tex]\( n(x) = m(x) + 5 \)[/tex]
- Adding 5 to [tex]\( m(x) \)[/tex] shifts the graph vertically upwards or downwards, which alters the intercepts and affects the graph's shape overall. This would change the [tex]\( x \)[/tex]-intercept location, which does not meet the required condition.
Option C: [tex]\( n(x) = m(x - 5) \)[/tex]
- This option represents a horizontal shift of the function [tex]\( m(x) \)[/tex] to the right by 5 units. To confirm, substituting [tex]\( x = 5 \)[/tex]:
[tex]\[ n(5) = m(5 - 5) = m(0) \][/tex]
For [tex]\( n(x) \)[/tex] to have an [tex]\( x \)[/tex]-intercept at [tex]\( x = 5 \)[/tex]:
[tex]\[ n(5) = 0 \implies m(0) = 0 \][/tex]
Therefore, the intercept at [tex]\( x = 5 \)[/tex] for [tex]\( n \)[/tex] would hold if [tex]\( m \)[/tex] originally had an intercept at [tex]\( x = 0 \)[/tex]. This transformation ensures the [tex]\( x \)[/tex]-intercept is set at [tex]\( x = 5 \)[/tex] appropriately.
Option D: [tex]\( n(x) = m(x + 5) \)[/tex]
- This option would move the function [tex]\( m(x) \)[/tex] left by 5 units. Substituting [tex]\( x = 5 \)[/tex]:
[tex]\[ n(5) = m(5 + 5) = m(10) \][/tex]
Which means that for [tex]\( n(x) \)[/tex] to be zero at [tex]\( x = 5 \)[/tex], [tex]\( m(10) \)[/tex] would need to be zero, which does not ensure that [tex]\( x = 5 \)[/tex] is the intercept in the context provided.
Considering these options, the correct answer is:
[tex]\[ \boxed{C} \][/tex]
This option correctly adjusts the input of [tex]\( m(x) \)[/tex] in a manner that ensures both the vertical asymptotes remain the same and the [tex]\( x \)[/tex]-intercept is specifically located at [tex]\( x = 5 \)[/tex].
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.
