Everyday usage and use in mathematics

In everyday English, the word domain describes a political region. The Oxford English Dictionary defines domain as "A district or region under rule, control, or influence, or contained within certain limits; realm; sphere of activity, influence, or dominion."[1] For example, the domain of King Charles III is the commonwealth of nations. The word domain is also used to describe a figurative area or region. For example, science and history are sometimes called domains of knowledge.

In mathematics, the word domain refers to a set of points where a function is defined. In other words, the domain of a function is the set of possible inputs to a function.

For example, consider the real-valued function:\[f(x) = \sqrt{x + 1}\]In order to make sense of \(\sqrt{x + 1}\) as a real number, it is necessary that the term \(x + 1\) occurring in the square root is at least zero. Hence \(x + 1 \geq 0\) and \(x \geq - 1\). So the domain of the function \(f(x)\) consists of those \(x\) such that \(x \geq -1\).

It is common in mathematics to talk about the inputs to a function as places, so the domain of a function is thought of as a region, agreeing with the everyday use of the term. For example, it is common to refer to \(f(x)\) as the value of the function \(f\) at the point \(x\). The set of numbers \(x\) satisfying the inequality \(x \geq -1\) is an infinite ray on the real number line. It is useful to think of this ray as the region where the expression \(f(x)\) makes sense.

The word domain is translated into Spanish as dominio. The everyday meaning of dominio in Spanish is similar to the English meaning in both its literal and figurative usage.

Summary

The word domain refers to a region and is commonly used both literally and figuratively. In mathematics, the domain of a function is the region on which the function is defined.

Importance in mathematics

In order to plug a value \(x\) into a function, it is important that \(x\) is inside of the domain of the function. Note that the domain of a function can be any set. The elements of the domain can be numbers, points, sets, functions, or anything else. Elements of the domain don't even need to be mathematical objects!

For example, if \(h(x)\) is a function that takes animals as inputs and gives the number of legs of the animal as an output, then worm, dog, and human are all elements of the domain of \(h\). We can plug these elements into h to get \(h(\text{worm}) = 0\), \(h(\text{dog}) = 4\), and \(h(\text{human}) = 2\).

In this example, It is important that the input \(x\) into the function \(h(x)\) is an animal. In this example, it may be helpful to think of the word "domain" in its figurative meaning.

Exercises

1. Describe the mathematical and everyday meaning of the word domain in your native language. Try to explain how the domain of a function is related to the everyday definition of domain as a region.
2. Describe in your own words how to find the domain of the function \(f(w) = \frac{1}{w^2 - 1}\).
3. Give an example of a function \(g(z)\) such that the domain of \(g\) consists of the days of the week.

References

[1] Oxford English Dictionary, s.v. “domain, n., sense 3.b”, September 2023. https://doi.org/10.1093/OED/8008562811