To understand the behavior and range of the exponential function [tex]\( k(x) \)[/tex], let's analyze the given information step-by-step:
1. Horizontal Asymptote:
We are told that the function [tex]\( k(x) \)[/tex] approaches a horizontal asymptote at [tex]\( y = 3 \)[/tex]. This means that as [tex]\( x \)[/tex] increases (towards positive infinity or negative infinity), the value of [tex]\( k(x) \)[/tex] will get closer and closer to 3 but never actually reach 3.
2. Exponential Growth:
The function [tex]\( k(x) \)[/tex] increases at a rate of [tex]\( 75 \% \)[/tex] and passes through the point [tex]\( (0, 9) \)[/tex]. An exponential function with this growth rate will grow rapidly away from its horizontal asymptote. Since it passes through [tex]\( (0, 9) \)[/tex], we know [tex]\( k(0) = 9 \)[/tex].
3. Behavior Based on the Starting Point and Asymptote:
- At [tex]\( x = 0 \)[/tex], [tex]\( k(x) = 9 \)[/tex].
- As [tex]\( x \)[/tex] increases, since [tex]\( k(x) \)[/tex] is an increasing function that approaches but does not cross the asymptote [tex]\( y = 3 \)[/tex], the values of [tex]\( k(x) \)[/tex] will get closer to 3 from above.
4. Range Determination:
Given that the function is always above the horizontal asymptote [tex]\( y = 3 \)[/tex] and it continues to increase without bound:
- The lowest possible value [tex]\( k(x) \)[/tex] can approach is [tex]\( y = 3 \)[/tex], but it will never actually reach 3.
- Thus, the values of [tex]\( k(x) \)[/tex] will be greater than 3.
- As [tex]\( x \)[/tex] continues to increase, [tex]\( k(x) \)[/tex] will also increase towards infinity.
Therefore, the range of the function [tex]\( k(x) \)[/tex] includes all [tex]\( y \)[/tex]-values greater than 3. We express this as the interval from 3 to infinity, excluding 3 itself.
Thus, the range of the function [tex]\( k(x) \)[/tex] is:
[tex]\[ (3, \infty) \][/tex]
So the correct answer is:
[tex]\[ (3, \infty) \][/tex]
