At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Discover in-depth solutions to your questions from a wide range of experts on our user-friendly Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
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]
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]
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.
