A reasonable domain contains the values of the independent variable for which the model is meaningful and sufficiently reliable in its real-world context. Determine it by considering physical restrictions, the data used to create the model, and any stated time or measurement limits.
How to determine the domain
Start by identifying the independent variable, usually represented on the horizontal axis. Then apply these checks:
| Check | Question to ask | Possible restriction |
|---|---|---|
| Context | Which input values are physically meaningful? | Time and distance usually satisfy . |
| Data range | Over which values was the model developed or fitted? | A model fitted for is most reliable within that interval. |
| Type of variable | Is the variable continuous or discrete? | A number of people must be a non-negative integer. |
| Practical limits | Does the situation impose a maximum or endpoint? | A concert venue may hold at most people. |
| Model behaviour | Does the model eventually produce impossible outputs? | A population model should not be used where it predicts a negative population. |
For example, suppose the height of a ball is modelled by
where is the time in seconds after the ball is thrown. The mathematical formula accepts every real value of , but negative times are not meaningful. Using a GDC to solve gives the positive root , when the ball reaches the ground. Therefore, a reasonable domain is
The common misconception is that the domain is automatically all values accepted by the formula. That is the mathematical domain; a modelling question requires the narrower contextual domain.
Exam technique
This is SL 2.6: Modelling skills, studied at both SL and HL and commonly tested in Paper 1 or Paper 2. State the interval, identify the variable and units, and justify both endpoints from the context. If predictions extend beyond the observed data, explicitly identify this as extrapolation and explain why reliability may decrease.
