# The Parking Lot

Alignments to Content Standards: F-IF.A.1

A parking lot charges $\$0.50$for each half hour or fraction thereof, up to a daily maximum of$\$10.00$. Let $C(t)$ be the cost in dollars of parking for $t$ minutes.

1. Complete the table below.

$t$ (minutes) $C(t)$ (dollars)
0
$15$
$20$
$35$
$75$
$125$
2. Sketch a graph of $C$ for $0 \leq t \leq 480$.
3. Is $C$ a function of $t$? Explain your reasoning.
4. Is $t$ a function of $C$? Explain your reasoning.

## IM Commentary

The purpose of this task is to investigate the meaning of the definition of function in a real-world context where the question of whether there is more than one output for a given input arises naturally. In more advanced courses this task could be used to investigate the question of whether a function has an inverse.

## Solution

1. $t$ (minutes) $C(t)$ (dollars)
0 0
$15$ $0.50$
$20$ $0.50$
$35$ $1.00$
$75$ $1.50$
$125$ $2.50$
As a sample calculation, we note that $125$ minutes is two full hours (four half-hours) and part of another half hour. Since the ticketing scheme rounds up to the nearest half-hour, we have to pay for five half-hours, at a total cost of \$2.50. 2. 3. Yes,$C$is a function of$t$because for a given parking time of$t$minutes there is exactly one charge. 4. No,$t$is not a function of$C$because there are values of$C$that have many values of$t$associated with them. For example if you end up paying$\$0.50$ then you could have parked for any period of time up to half an hour, that is, when $C = 0.50$ then $t$ can have any value in the range $0 < t \le 30$. So the "input" $C = 0.50$ yields more than one output, which is not allowed for a function.