Find a formula for B(d).

Question

Your internet service provider charges you a different monthly bill, B (in dollars), depending on how much data you use, d (in gigabytes). The base amount of $65 includes up to 50 gigabytes of data. For every gigabyte of data above 50, you are charged $2, up to a maximum bill of $165. Find a formula for B(d).

Answer 1

This one is a bit tricky. The function can't be expressed simply as a single equation. The function is what is known as a piece-wise defined function.

We are told that: "The base amount of $65 includes up to 50 gigabytes of data.".

This would mean that for any value of d between 0 and 50, the cost B(d) is $65.

Then for every gigabyte above 50, we will start to be charged 2 dollars. The number of gigabytes above 50 is d-50. So then the formula which applies will be B(d) = 65 + 2 (d-50)

But then we are told the maximum charge is 165 dollars. But it is not immediately clear how many gigabytes can we use before that happens. So what we will do is solve the equation

B(d) = 165 = 65 + 2 (d-50) subtract 65 from both sides

100 = 2(d-50) divide by 2

50 = d-50 add 50

100 = d gigabytes

This is when the price reaches 165 dollars

So putting all this together

B(d) = { 50 if 0≤d≤50, 65 + 2 (d-50) if 50≤d≤100 , and 165 if d≥100 }

Answer 2

You could write this as a piecewise function like this:

Where d is data in gigabytes and B is price in dollars.

So, for d up to and including 50 GB, you use the top rule (B = $65), for d between 50 and 100 GB you use the middle rule (for example if d = 60, the bill should be $65 plus 10•2 or $85 and the function 2(60-50)+65 would give you that). And for data over 100 GB, you pay $165. To find that upper limit of 100 GB, you need to set the middle rule equal to 165 and solve for d.