Domain of a Function

The Domain of a Function is a term in mathematics referring to all of the $x$ values where a function is defined. In more technical language, it is the mathematical set of all inputs where the function is defined.

One way to express this mathematically is $f:A \to B$ or $f:X \to Y$, where the “function that input of set A maps to output set B” or the “function where input of set X maps to output of set Y”. In both cases $x$ is part of set A or X, while $y$ is a member of set B or Y.

(The uppercase “X” or “Y”refers to a set — versus the members of those sets, $x$ and $y$ respectively. It is the same if we choose to use designate them as set A and B instead.)

We often graph functions in a Cartesian Coordinate System, a special case where A and B are sets of Real Numbers.

Natural Domain

There are many functions where the natural domain is restricted by its very nature. Take, for example, the function

\[ f(x) = \sqrt{x} \]

The Square Root Function is a unique type of function as it cannot be evaluated if $x$ is a negative number — the square root of a negative number does not exist because if we are looking for a number that multiples with itself to become $x$ we will never have a negative result as a negative number multiplied by itself becomes positive. Thus we cannot take the square root of a negative number.

Therefore the natural domain of a square root function is the set of all non-negative real numbers (including zero and all positive integers). Thus its domain is everything from zero to positive infinity. The domain of $f(x)=\sqrt{x}$ is the interval $[0, \infty)$ or $X \setof R$ where $x \geq 0$.

Domain Restrictions, Partial Functions, and Piecewise Functions

Any function can be restricted to a fraction of its domain, which we refer to as a “subset” of its domain. In physics (and the natural sciences in general) this is relevant in certain circumstances because it doesn’t make sense for a function to be defined on the entirety of its domain.

Above we saw how the natural domain of a function can be by nature a subset of real numbers (as in the Square Root Function). In science we often impose our own restrictions. Consider the case if we are working with the function describing the distance traveled by a falling body:

\[ d = \frac{1}{2}gt^2 \]

Where $d$ is the distance traveled (in meters), $g$ is the gravitational constant ($9.81 \frac{m^2}\{s^2}$) and $t$ is the time (in seconds). Under no normal circumstances does it make sense for us to consider a negative value for time, $t$. Thus if we graph our values for this equation, our input is $t$ which maps to the x-axis, while the output is is distance $d$ which maps to the y-axis.

Thus our domain is the set of real numbers, $x \setof R$ where $x \geq 0$. This is a restriction we place on the function because of the influence of reality on the math.

A better example would be if we write a function that relates year (T) to the total human population (P) — $f: T \to P$ — which has a very specific start time of the evolution of humans ~300,000 BCE. It makes sense to consider a domain before that because the population will always be zero.

Notes

Resources

1. “Domain of a Function.” Wikipedia. <https://en.wikipedia.org/wiki/Domain_of_a_function>

[latexpage]