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Definition of Domain and Range
What does domain mean in algebra? Many students struggle with remembering the difference between the domain and the range. To start, let's back up and define a function and a relation.
A relation is a mathematical object that maps elements from one set, called the domain, to another set (which could be the same set) called the range. A function is a special type of relation, where every element of the domain maps to exactly one element of the range.
When a relation is drawn as a graph, the domain is always on the x-axis and the range is always on the y-axis. The vertical line test for a function is simply this: if any vertical line intersects the graph of the relation more than once, it fails to be a function because the same element of the domain (x-value) is mapped to several elements of the range (y-value).
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Examples of a Function
Consider the process of squaring a number. This is a function, because each and every number has exactly one square. On the other hand, taking the square root is not a function, as a number can have several square roots (although it has only one principal square root).
Suppose we take the domain of a squaring function to be all real numbers; then the range is all nonnegative numbers, because zero squared is zero and every other real number squared is a positive number.
Another simple function would be f(x) = f(x)+3. In this case, if the domain is all real numbers, the range is also all real numbers; these are the same set. This function is described as one to one, because every element in the second set is paired with no more than one element in the first set. It is called onto because every element in the second set is paired with at least one element in the first set. Being both one to one and onto makes the function bijective.
Notice that the elements do not need to be numbers! In the previous example, the domain and range could both be the English alphabet, and the function would return the letter that appears three spaces after the input, with the later letters wrapping around (that is, Z would become C). This is called a Caesar Cipher.
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Summing Up Domain
So, what does domain mean in algebra? Domain is simply the set of elements (which could be numbers or other values) which are used as input for a relation. The same relation can be applied to multiple domains.
We also say that the domain is the independent variable and the range is the dependent variable, because we can often choose any numbers we want for the domain and plug them into a function, which will return values from the range; in this case, which elements belong to the range depends on the elements chosen for the domain.