Answer :
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].
