Modeling Real-World Relationships with Functions

From a relationship to a function

Suppose a tank contains 18 litres of water and a pump adds water at a constant rate of 4 litres per minute. If $t$ is the number of minutes since the pump started, the volume in the tank is

Read $V(t)$ as “$V$ of $t$”: the volume determined by the time $t$. The number 18 is the starting volume, and $4t$ is the volume added after $t$ minutes. For example,

In words, 2.5 minutes after the pump starts, the model gives a volume of 28 litres. The input $t$ has units of minutes; the output $V(t)$ has units of litres. Keeping those units beside the variables makes the notation meaningful rather than just symbolic.

A function assigns one output to each permitted input. That does not mean different inputs must have different outputs. It means that once an allowed input is specified, the rule gives an unambiguous result. The letter naming the function can reflect what its output represents: $V$ for volume or $C$ for cost, for instance. The parentheses in $V(t)$ indicate an input, not multiplication.

The situation determines which inputs are allowed

An algebraic formula and a real-world model are not quite the same thing. You can calculate $18+4t$ for $t=-3$, but “three minutes since the pump started” cannot be negative. More subtly, you can calculate it for $t=100$, yet that does not tell you whether the pump is still running then.

The domain is the set of permitted inputs. To choose it, ask two questions:

1. Does the formula give a mathematically defined output for this input?
2. Does the input—and the use of this formula at that input—make sense in the stated situation?

The first question catches issues such as division by zero. The second catches negative times, fractional counts, capacity limits and times beyond the period a process is being modelled.

A price of 1.50 per unit tells you how to calculate a cost; it does not tell you how much stock is available. A maximum purchase of 100 units is justified only if we add an assumption such as “the shop can sell at most 100 units.” Do not invent a convenient bound and present it as though it were given.

Now suppose our tank’s pump runs for exactly 7 minutes. Time can be measured to fractions of a minute, so an appropriate model is

where $t$ is measured in minutes and $V(t)$ in litres. The domain is the real numbers from 0 through 7, including both endpoints. At $t=0$, the tank contains 18 litres; at $t=7$, it contains $V(7)=46$ litres. Calculating $V(8)=50$ is possible algebraically, but it is not a prediction licensed by this model: the stated pumping period has ended.

When inputs are counts rather than measurements

Time in the tank example can take values such as $2.5$ minutes. An order quantity behaves differently. Suppose a supplier charges a fixed delivery fee of 14 currency units plus 6 currency units per laboratory kit, and accepts orders of 1 to 30 whole kits. Let $n$ be the number of kits and $C(n)$ the total cost. Then

Here $C(3)=32$ currency units: the fixed fee plus the cost of three kits. The formula also produces an answer for $n=2.5$, but an order for two and a half whole kits is not allowed. It produces an answer for $n=0$, but the supplier does not accept a zero-kit order. Neither input belongs to the applied domain, even though both can be substituted into the expression $14+6n$.

Inputs like these form a discrete set: separate allowed values rather than every value between two endpoints. Counts are often discrete, while measurements such as elapsed time or mass are often treated as continuous. The context still has the final say—a time recorded only in whole-minute steps, for example, would have a different domain from continuously measured time.

State the domain so its boundaries are unmistakable

For a continuous input, an inequality or interval notation can express the same domain. The tank model’s domain can be written either as $0\le t\le 7$ or as $t\in [0,7]$. Square brackets mean the endpoints are included; round brackets mean they are excluded.

For example, if a process is defined only after time 0 but through time 7, its continuous domain is $0<t\le 7$, written $t\in (0,7]$. The choice between an open and a closed endpoint should follow the wording of the situation.

For the kit order, however, $[1,30]$ would be misleading: it includes $1.5$, $2.5$ and every other real number between 1 and 30. The set $\{1,2,\dots ,30\}$ states the intended inputs. Equivalently, you could write $1\le n\le 30$ with $n$ an integer.

A useful final check is to read your finished model aloud. Does the input variable name a quantity and unit? Does the output name a quantity and unit? Do the smallest, largest and in-between inputs in your stated domain all make sense? If a boundary is absent from the question, state the assumption you need rather than silently adding it.

Conclusion

A real-world function consists of more than a calculation rule. Function notation identifies which quantity depends on which input, while the domain identifies the inputs for which the rule is intended. Check both mathematical validity and the situation’s constraints; distinguish continuous measurements from discrete counts; and state any assumed limits explicitly.

In the next lesson, you will solve quadratic equations using the quadratic formula. The habit established here—checking what an answer means in context—will remain important when an equation produces values that are mathematically valid but unsuitable for a particular model.