Let's analyze the function [tex]\( C(t) = 3(2)^t \)[/tex] step-by-step to evaluate the given statements.
### Breakdown of the Function:
- Initial Value: When [tex]\( t = 0 \)[/tex],
[tex]\[
C(0) = 3(2)^0 = 3 \times 1 = 3
\][/tex]
This means that Nick started his business with 3 clients.
- Growth Factor: The function [tex]\( C(t) = 3(2)^t \)[/tex] shows exponential growth where the growth factor is determined by the base of the exponent, which is 2. This means the number of clients doubles every month.
Now, let’s evaluate each statement given in the problem:
1. "The growth rate is 3, which means each month, Nick has 3 times as many clients as he had the month before."
- This statement is incorrect. The coefficient 3 represents the initial number of clients, not the growth rate. The growth rate or factor is 2, meaning the number of clients doubles each month, not triples.
2. "The growth rate of Nick's clients is [tex]\(2\% \)[/tex] per month."
- This statement is incorrect. A 2% growth rate would mean the clients increase by 2% every month. However, in the function [tex]\( C(t) = 3(2)^t \)[/tex], the clients double (100% increase), not merely increase by 2%.
3. "The initial value is 2, which means that Nick started his business with 2 clients."
- This statement is incorrect. As calculated, the initial value (when [tex]\( t = 0 \)[/tex]) is 3, not 2. Nick started his business with 3 clients.
4. "The horizontal asymptote is [tex]\(y=0\)[/tex], but it does not mean anything in this situation because Nick starts his company with 3 clients."
- This statement is somewhat true. Let's elaborate:
- The horizontal asymptote for the function [tex]\( C(t) = 3(2)^t \)[/tex] is [tex]\( y = 0 \)[/tex]. This is a theoretical value indicating that as [tex]\( t \)[/tex] becomes very large, the number of clients [tex]\( C(t) \)[/tex] grows exponentially and will never approach zero.
- However, mentioning the horizontal asymptote [tex]\( y = 0 \)[/tex] in the context of this application is not entirely relevant, because it does not provide useful information about the business's client growth. Nick's business indeed starts with 3 clients and grows exponentially from there.
### Conclusion:
The only somewhat accurate statement when interpreting the exponential function [tex]\( C(t) = 3(2)^t \)[/tex] is:
>"The horizontal asymptote is [tex]\(y = 0\)[/tex], but it does not mean anything in this situation because Nick starts his company with 3 clients."
