What would be the function for this?

Question

A cell phone plan has $700$ minutes of talking for $20$ dollars with each additional minute over $700$ minutes costing $10$ cents per minute.

I had $f(x)=0.1t-50$ but then I plugged in $0$ and got $-50$ which is wrong.

I'm pretty confused right now.

Answer 1

It's a conditional function:

t = minutes in whole minutes.

f(t) = 20 if t <= 700;

f(t) = .1*t - 50; if t > 700

======

If you want to get technical. Let floor(t) = the largest integer equal or less than t. (This is a stepwise non-continuous function).

f(t) = 20 if t<=700.

f(t) = .1*floor(t) - 50; if t > 700.

Answer 2

You could use a piecewise-defined function. Or you could be cooler and use only one equation:

$$f(x) = \frac{\lceil x\rceil +\left|\lceil x \rceil -700\right|}{20}-15$$

This uses the ceiling function for rounding up to the minute. The vertical bars denote absolute value.

Answer 3

Probably the most concise version of your function will be $$ f(t)= \begin{cases} 20 & \text{if $0<t\leq 700$},\\ 20 + 0.1(\lceil t\rceil - 700) & \text{if $t>700$}. \end{cases} $$ Does that make sense?

Note: The notation $\lceil t\rceil$ stands for the ceiling function.