Let's break down and address the questions from the image:

(a) Find a linear function that expresses the monthly bill $B(m)$ as a function of minutes used $m$: The linear function given in the question is:

$B(m) = 0.01m + 5.17$

This equation means that the monthly bill $B(m)$ is $5.17 plus $0.01 per minute used.

(b) What are the independent and dependent variables?

• The independent variable is $m$, which represents the number of minutes used.
• The dependent variable is $B(m)$, which represents the monthly bill.

(c) What is the implied domain of this linear function for a 31-day month?

• The number of minutes used in a 31-day month can range from 0 minutes to the maximum possible minutes in 31 days.
• Maximum number of minutes in 31 days = $31 \times 24 \times 60 = 44,640$ minutes.
• Therefore, the domain is $[0, 44,640]$.

(d) What is the monthly bill if 1,297 minutes are used for long-distance calls?

• Plug $m = 1,297$ into the equation $B(m) = 0.01m + 5.17$:

$B(1,297) = 0.01(1,297) + 5.17 = 12.97 + 5.17 = 18.14$

The monthly bill would be $18.14.

If you'd like to dive deeper into any of these steps, let me know! Here are five questions to consider further:

1. What does the slope (0.01) represent in the linear equation?
2. How would the equation change if the cost per minute was $0.05 instead?
3. What would the monthly bill be if no minutes were used?
4. How would the maximum possible bill change for a month with 30 days?
5. How can we graph this function, and what would the key points look like?

Tip: In linear equations, the slope is often a rate of change, showing how the dependent variable changes with respect to the independent variable.