Answer :
To determine the range of the exponential function [tex]\( k(x) \)[/tex], let's carefully analyze its behavior, attributes, and characteristics based on the information given.
### Key Points
1. Exponential Growth and Ordered Pair:
- The function [tex]\( k(x) \)[/tex] increases at a rate of [tex]\(75\%\)[/tex], implying it is an exponential growth function.
- Given the ordered pair [tex]\((0, 8)\)[/tex], this indicates that when [tex]\( x = 0 \)[/tex], [tex]\( k(0) = 8 \)[/tex].
2. Horizontal Asymptote:
- The function approaches a horizontal asymptote at [tex]\( y = 2 \)[/tex]. This means as [tex]\( x \)[/tex] approaches positive or negative infinity, the function value [tex]\( k(x) \)[/tex] gets closer and closer to [tex]\( 2 \)[/tex], but never actually reaches it.
### Characteristics of the Exponential Function [tex]\( k(x) \)[/tex]:
- For an exponential function of the form [tex]\( k(x) = A \cdot b^x + C \)[/tex], where [tex]\( A \)[/tex], [tex]\( b \)[/tex], and [tex]\( C \)[/tex] are constants:
- [tex]\( A \)[/tex] is a coefficient that determines the vertical stretch or compression and affects the initial value.
- [tex]\( b \)[/tex] is the base of the exponential function.
- [tex]\( C \)[/tex] is the horizontal asymptote, which in this case is [tex]\( y = 2 \)[/tex].
Given that as [tex]\( x \rightarrow \infty \)[/tex] (or [tex]\( x \rightarrow -\infty \)[/tex]), [tex]\( k(x) \)[/tex] approaches 2, we can represent the function in the form:
[tex]\[ k(x) = A \cdot b^x + 2 \][/tex]
Since [tex]\( k(0) = 8 \)[/tex]:
[tex]\[ 8 = A \cdot b^0 + 2 \][/tex]
[tex]\[ 8 = A \cdot 1 + 2 \][/tex]
[tex]\[ A + 2 = 8 \][/tex]
[tex]\[ A = 6 \][/tex]
Thus, our function can be written more specifically as:
[tex]\[ k(x) = 6 \cdot b^x + 2 \][/tex]
Where [tex]\( b \)[/tex] indicates the growth rate, which corresponds to a [tex]\( 75\% \)[/tex] increase.
### The Range of [tex]\( k(x) \)[/tex]:
- The exponential growth function will keep increasing without bound as [tex]\( x \)[/tex] increases, subtracting the horizontal asymptote value [tex]\( 2 \)[/tex], indicating we are adding [tex]\( 6 \cdot b^x \)[/tex] to [tex]\( 2 \)[/tex]:
- As [tex]\( x \to \infty \)[/tex], [tex]\( k(x) \to \infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( k(x) \)[/tex] approaches [tex]\( y = 2 \)[/tex], but never dips below, since the exponential part [tex]\( 6 \cdot b^x \)[/tex] decreases towards [tex]\(0\)[/tex], making the function approach [tex]\(2\)[/tex] from above.
Hence, [tex]\( k(x) \)[/tex] values will never be less than [tex]\(2\)[/tex].
### Conclusion:
Given [tex]\( k(x) \)[/tex] always stays above [tex]\( y = 2 \)[/tex] and increases without bound:
The range of the function [tex]\( k(x) \)[/tex] is [tex]\( (2, \infty) \)[/tex].
So, the correct answer is:
[tex]\[ \boxed{(2, \infty)} \][/tex]
### Key Points
1. Exponential Growth and Ordered Pair:
- The function [tex]\( k(x) \)[/tex] increases at a rate of [tex]\(75\%\)[/tex], implying it is an exponential growth function.
- Given the ordered pair [tex]\((0, 8)\)[/tex], this indicates that when [tex]\( x = 0 \)[/tex], [tex]\( k(0) = 8 \)[/tex].
2. Horizontal Asymptote:
- The function approaches a horizontal asymptote at [tex]\( y = 2 \)[/tex]. This means as [tex]\( x \)[/tex] approaches positive or negative infinity, the function value [tex]\( k(x) \)[/tex] gets closer and closer to [tex]\( 2 \)[/tex], but never actually reaches it.
### Characteristics of the Exponential Function [tex]\( k(x) \)[/tex]:
- For an exponential function of the form [tex]\( k(x) = A \cdot b^x + C \)[/tex], where [tex]\( A \)[/tex], [tex]\( b \)[/tex], and [tex]\( C \)[/tex] are constants:
- [tex]\( A \)[/tex] is a coefficient that determines the vertical stretch or compression and affects the initial value.
- [tex]\( b \)[/tex] is the base of the exponential function.
- [tex]\( C \)[/tex] is the horizontal asymptote, which in this case is [tex]\( y = 2 \)[/tex].
Given that as [tex]\( x \rightarrow \infty \)[/tex] (or [tex]\( x \rightarrow -\infty \)[/tex]), [tex]\( k(x) \)[/tex] approaches 2, we can represent the function in the form:
[tex]\[ k(x) = A \cdot b^x + 2 \][/tex]
Since [tex]\( k(0) = 8 \)[/tex]:
[tex]\[ 8 = A \cdot b^0 + 2 \][/tex]
[tex]\[ 8 = A \cdot 1 + 2 \][/tex]
[tex]\[ A + 2 = 8 \][/tex]
[tex]\[ A = 6 \][/tex]
Thus, our function can be written more specifically as:
[tex]\[ k(x) = 6 \cdot b^x + 2 \][/tex]
Where [tex]\( b \)[/tex] indicates the growth rate, which corresponds to a [tex]\( 75\% \)[/tex] increase.
### The Range of [tex]\( k(x) \)[/tex]:
- The exponential growth function will keep increasing without bound as [tex]\( x \)[/tex] increases, subtracting the horizontal asymptote value [tex]\( 2 \)[/tex], indicating we are adding [tex]\( 6 \cdot b^x \)[/tex] to [tex]\( 2 \)[/tex]:
- As [tex]\( x \to \infty \)[/tex], [tex]\( k(x) \to \infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( k(x) \)[/tex] approaches [tex]\( y = 2 \)[/tex], but never dips below, since the exponential part [tex]\( 6 \cdot b^x \)[/tex] decreases towards [tex]\(0\)[/tex], making the function approach [tex]\(2\)[/tex] from above.
Hence, [tex]\( k(x) \)[/tex] values will never be less than [tex]\(2\)[/tex].
### Conclusion:
Given [tex]\( k(x) \)[/tex] always stays above [tex]\( y = 2 \)[/tex] and increases without bound:
The range of the function [tex]\( k(x) \)[/tex] is [tex]\( (2, \infty) \)[/tex].
So, the correct answer is:
[tex]\[ \boxed{(2, \infty)} \][/tex]
