A domain is a set of all inputs (e.g., all *x*-values). In many cases, the domain is the set of all real numbers. However, sometimes a problem calls for a contextual domain — a narrower domain based on context. In other words, it’s the values between two numbers that *x* can take on based on a particular situation.

## Contextual Domain: Example

**Example question 1**: You want to create a box with a surface area of 900 cm^{2}. What is the contextual domain [1]?

**Solution**:

Step 1: **Write an inequality** for the problem. For this example, *x* can be from 0 to 1000 cm^{2}, which gives the inequality

0 < x^{2} < 900.

Step 2: **Solve for x.** For this example, we can find

*x*by taking the square root of all terms.

√0 < √(x

^{2}) < √900 →

0 < x < 300.

The contextual domain is usually informational and not connected to the problem’s structure or the math needed to solve the problem [2]. For example, you don’t need to know the contextual domain to work with probability distributions, nor do you need to know the fact that a domain is a specific number, say (0, 300), to solve the problem (although it certainly helps).

**Example question 2**: A shipping company owns 1000 shipping containers. You want to find how many bags of flour can fit into *x* full containers. If the formula for the number of bags of flour that can fit into one shipping container is f(x) = 3000x, what is the contextual domain?

**Solution**: we have x shipping containers here. As we are given the condition that the shipping containers must be full, that means we can’t have halves, quarters, or any other fractions of shipping containers. We also can’t have negative numbers, as this would make no sense in the context of the problem. As we have a maximum of 1,000 shipping containers available, the contextual domain is the set of positive integers {1, 2, …, 1000).

## References

[1] Simmonds (2011). MTH 252 Test 1. Retrieved November 11, 2021 from: http://spot.pcc.edu/~ssimonds/mth_252/2016_test_2_pratice/MTH_252_simonds_key_2_practice_201601_11.pdf

[2] Third Misconceptions Seminar Proceedings: Students Misconceptions and Errors in Solving Algebra Word Problems Related to Misconceptions in the Field of Science (2011) from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.800.5569&rep=rep1&type=pdf