At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Experience the ease of finding quick and accurate answers to your questions from professionals on our platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
Let's analyze each of the given functions step-by-step to determine their horizontal and vertical asymptotes.
### Function [tex]\( f(x) = 5000(0.4)^x \)[/tex]
This function represents exponential decay because the base of the exponent, 0.4, is between 0 and 1. As [tex]\( x \)[/tex] approaches infinity, [tex]\( 0.4^x \)[/tex] approaches 0. Therefore, the function [tex]\( f(x) = 5000(0.4)^x \)[/tex] will tend to 0 as [tex]\( x \)[/tex] becomes very large.
Horizontal Asymptote:
- Given that [tex]\( f(x) \)[/tex] approaches 0 as [tex]\( x \)[/tex] approaches infinity, the horizontal asymptote of this function is [tex]\( y = 0 \)[/tex].
### Function [tex]\( f(x) = 5000(0.6)^x \)[/tex]
Similar to the previous function, this one also represents exponential decay because the base of the exponent, 0.6, is between 0 and 1. As [tex]\( x \)[/tex] approaches infinity, [tex]\( 0.6^x \)[/tex] approaches 0. Therefore, the function [tex]\( f(x) = 5000(0.6)^x \)[/tex] will also tend to 0 as [tex]\( x \)[/tex] becomes very large.
Horizontal Asymptote:
- Given that [tex]\( f(x) \)[/tex] approaches 0 as [tex]\( x \)[/tex] approaches infinity, the horizontal asymptote of this function is [tex]\( y = 0 \)[/tex].
### Function [tex]\( f(x) = 5000(1.4)^x \)[/tex]
This function represents exponential growth because the base of the exponent, 1.4, is greater than 1. As [tex]\( x \)[/tex] approaches infinity, [tex]\( 1.4^x \)[/tex] grows unbounded, meaning it increases without limit.
Horizontal Asymptote:
- As [tex]\( f(x) \)[/tex] grows unboundedly, it does not approach any finite value. Therefore, it does not have a horizontal asymptote at [tex]\( y = 0 \)[/tex].
Vertical Asymptote:
- Since [tex]\( f(x) \)[/tex] is defined for all real values of [tex]\( x \)[/tex] and grows smoothly without approaching infinity or negative infinity for any specific [tex]\( x \)[/tex], it does not have a vertical asymptote at [tex]\( x = 0 \)[/tex].
### Function [tex]\( f(x) = 5000(1.6)^x \)[/tex]
This function also represents exponential growth because the base of the exponent, 1.6, is greater than 1. As [tex]\( x \)[/tex] approaches infinity, [tex]\( 1.6^x \)[/tex] grows unbounded.
Horizontal Asymptote:
- As [tex]\( f(x) \)[/tex] grows unboundedly, it does not approach any finite value. Therefore, it does not have a horizontal asymptote at [tex]\( y = 0 \)[/tex].
Vertical Asymptote:
- Since [tex]\( f(x) \)[/tex] is defined for all real values of [tex]\( x \)[/tex] and grows smoothly without approaching any finite value for a specific [tex]\( x \)[/tex], it does not have a vertical asymptote at [tex]\( x = 0 \)[/tex].
### Summary
- The function [tex]\( f(x) = 5000(0.4)^x \)[/tex] has a horizontal asymptote at [tex]\( y = 0 \)[/tex].
- The function [tex]\( f(x) = 5000(0.6)^x \)[/tex] has a horizontal asymptote at [tex]\( y = 0 \)[/tex].
- The function [tex]\( f(x) = 5000(1.4)^x \)[/tex] does not have a horizontal or vertical asymptote.
- The function [tex]\( f(x) = 5000(1.6)^x \)[/tex] does not have a horizontal or vertical asymptote.
Given these findings, the horizontal and vertical asymptote properties for the functions are as follows:
1. [tex]\( f(x) = 5000(0.4)^x \)[/tex]: Horizontal asymptote at [tex]\( y = 0 \)[/tex]
2. [tex]\( f(x) = 5000(0.6)^x \)[/tex]: Horizontal asymptote at [tex]\( y = 0 \)[/tex]
3. [tex]\( f(x) = 5000(1.4)^x \)[/tex]: No horizontal or vertical asymptotes.
4. [tex]\( f(x) = 5000(1.6)^x \)[/tex]: No horizontal or vertical asymptotes.
These results match the outcome: [True, True, False, False, False, False].
### Function [tex]\( f(x) = 5000(0.4)^x \)[/tex]
This function represents exponential decay because the base of the exponent, 0.4, is between 0 and 1. As [tex]\( x \)[/tex] approaches infinity, [tex]\( 0.4^x \)[/tex] approaches 0. Therefore, the function [tex]\( f(x) = 5000(0.4)^x \)[/tex] will tend to 0 as [tex]\( x \)[/tex] becomes very large.
Horizontal Asymptote:
- Given that [tex]\( f(x) \)[/tex] approaches 0 as [tex]\( x \)[/tex] approaches infinity, the horizontal asymptote of this function is [tex]\( y = 0 \)[/tex].
### Function [tex]\( f(x) = 5000(0.6)^x \)[/tex]
Similar to the previous function, this one also represents exponential decay because the base of the exponent, 0.6, is between 0 and 1. As [tex]\( x \)[/tex] approaches infinity, [tex]\( 0.6^x \)[/tex] approaches 0. Therefore, the function [tex]\( f(x) = 5000(0.6)^x \)[/tex] will also tend to 0 as [tex]\( x \)[/tex] becomes very large.
Horizontal Asymptote:
- Given that [tex]\( f(x) \)[/tex] approaches 0 as [tex]\( x \)[/tex] approaches infinity, the horizontal asymptote of this function is [tex]\( y = 0 \)[/tex].
### Function [tex]\( f(x) = 5000(1.4)^x \)[/tex]
This function represents exponential growth because the base of the exponent, 1.4, is greater than 1. As [tex]\( x \)[/tex] approaches infinity, [tex]\( 1.4^x \)[/tex] grows unbounded, meaning it increases without limit.
Horizontal Asymptote:
- As [tex]\( f(x) \)[/tex] grows unboundedly, it does not approach any finite value. Therefore, it does not have a horizontal asymptote at [tex]\( y = 0 \)[/tex].
Vertical Asymptote:
- Since [tex]\( f(x) \)[/tex] is defined for all real values of [tex]\( x \)[/tex] and grows smoothly without approaching infinity or negative infinity for any specific [tex]\( x \)[/tex], it does not have a vertical asymptote at [tex]\( x = 0 \)[/tex].
### Function [tex]\( f(x) = 5000(1.6)^x \)[/tex]
This function also represents exponential growth because the base of the exponent, 1.6, is greater than 1. As [tex]\( x \)[/tex] approaches infinity, [tex]\( 1.6^x \)[/tex] grows unbounded.
Horizontal Asymptote:
- As [tex]\( f(x) \)[/tex] grows unboundedly, it does not approach any finite value. Therefore, it does not have a horizontal asymptote at [tex]\( y = 0 \)[/tex].
Vertical Asymptote:
- Since [tex]\( f(x) \)[/tex] is defined for all real values of [tex]\( x \)[/tex] and grows smoothly without approaching any finite value for a specific [tex]\( x \)[/tex], it does not have a vertical asymptote at [tex]\( x = 0 \)[/tex].
### Summary
- The function [tex]\( f(x) = 5000(0.4)^x \)[/tex] has a horizontal asymptote at [tex]\( y = 0 \)[/tex].
- The function [tex]\( f(x) = 5000(0.6)^x \)[/tex] has a horizontal asymptote at [tex]\( y = 0 \)[/tex].
- The function [tex]\( f(x) = 5000(1.4)^x \)[/tex] does not have a horizontal or vertical asymptote.
- The function [tex]\( f(x) = 5000(1.6)^x \)[/tex] does not have a horizontal or vertical asymptote.
Given these findings, the horizontal and vertical asymptote properties for the functions are as follows:
1. [tex]\( f(x) = 5000(0.4)^x \)[/tex]: Horizontal asymptote at [tex]\( y = 0 \)[/tex]
2. [tex]\( f(x) = 5000(0.6)^x \)[/tex]: Horizontal asymptote at [tex]\( y = 0 \)[/tex]
3. [tex]\( f(x) = 5000(1.4)^x \)[/tex]: No horizontal or vertical asymptotes.
4. [tex]\( f(x) = 5000(1.6)^x \)[/tex]: No horizontal or vertical asymptotes.
These results match the outcome: [True, True, False, False, False, False].
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.
