Let's examine the problem step-by-step to determine the range of the total cost function \( N(t) \).
Step 1: Understand the function
The total cost \( N(t) \) for a member after \( t \) months is given by the linear function:
[tex]\[ N(t) = 3.75t + 20.25 \][/tex]
where \( t \) is the number of months.
Step 2: Analyze the components of the function
- The initial fee is [tex]$\$[/tex] 20.25$.
- The monthly membership fee is [tex]$\$[/tex] 3.75$ per month.
Step 3: Evaluate the function at the starting point
When \( t = 0 \) (i.e., right when the member joins and has not been charged for any additional months yet), the total cost is:
[tex]\[ N(0) = 3.75(0) + 20.25 = 20.25 \][/tex]
So, the minimum cost (when \( t = 0 \)) is [tex]$\$[/tex] 20.25$.
Step 4: Consider the behavior of the function as time increases
As \( t \) (the number of months) increases, the value of \( N(t) \) will increase linearly because of the positive coefficient \( 3.75 \). There is no upper limit to the number of months \( t \) can be since \( t \) can theoretically go to infinity.
So, as \( t \to \infty \):
[tex]\[ N(t) \to \infty \][/tex]
Step 5: Determine the range of the function
The minimum value of \( N(t) \) is \( 20.25 \) (when \( t = 0 \)). The maximum value of \( N(t) \) is unbounded (approaches infinity).
Hence, the range of the function \( N(t) = 3.75t + 20.25 \) is:
[tex]\[ [20.25, \infty) \][/tex]
Therefore, the correct answer is:
[tex]\[ [20.25, \infty) \][/tex]
