Question

a cell phone company charges $300 for a new phone and then charges you $40 a month, m, for service regardless of how many minutes you use. use t(m) = $300 + $40m to model the total bill, t. assume you are budgeting for 2 to 6 months.
what variable is the domain?
what is a reasonable domain for this situation?
what variable is the range?
what is a reasonable range for this situation?

Explanation:

Step1: Identify domain variable

The function is $T(m)=300 + 40m$, where $m$ represents the number of months. The domain is the set of input values, so the domain variable is $m$ (months).

Step2: Determine reasonable domain

We are budgeting for 2 to 6 months. So the reasonable domain is the set of integers from 2 to 6, i.e., $\{2,3,4,5,6\}$.

Step3: Identify range variable

The range is the set of output values. The function $T(m)$ gives the total bill $T$. So the range variable is $T$ (total bill).

Step4: Calculate reasonable range

When $m = 2$, $T(2)=300+40\times2=300 + 80=380$.
When $m = 3$, $T(3)=300+40\times3=300 + 120=420$.
When $m = 4$, $T(4)=300+40\times4=300 + 160=460$.
When $m = 5$, $T(5)=300+40\times5=300 + 200=500$.
When $m = 6$, $T(6)=300+40\times6=300 + 240=540$.
So the reasonable range is $\{380,420,460,500,540\}$.

Answer:

What variable is the domain? : $m$, months
What is a reasonable domain for this situation? : $\{2,3,4,5,6\}$
What variable is the range? : $T$, total bill
What is a reasonable range for this situation? : $\{380,420,460,500,540\}$