What Is the Definition of the Domain of Function?
Domain is defined by Merriam-Webster as the complete ownership of land or a territory, or a region demarcated by certain features. Domain also denotes a region of knowledge, activity or influence. The word "domain" comes from the Latin "dominium," which is from "dominus." This word means, dominium or kingdom.
In mathematics, the meaning of domain is very close to its original meaning. It is the set of numbers or other elements that define, or make up, a function. In other words, the domain is the territory that belongs to the function.
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Identification
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The domain of a function is the set of numbers that are fed into the function. The group of numbers that exit the function is the "range" of that function.
For example, the numbers of a domain will be represented by "x." These are the numbers that are put into the function, and they represent the independent variable. On the other hand, the numbers of the range of a function will represent "y." These are the numbers that result from the function, and they represent the dependent variables. Therefore, these numbers are dependent upon the "x" value.
Let's look at the effect of x (the domain) on the value of y (the range). If given this equation, as an example of a function:
y=4x
The "x" is the independent variable, which is a number chosen from within the domain of the function. So, if I choose the number 5 to substitute for x:
y=4(5), then
y=20
The resulting value for "y" equals 20. This "y" is the dependent variable, dependent upon whatever the value of x is. This is referred to as the range.
Significance
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The domain of a function must be specified to yield the desired results. Sometimes limits are put on the types of numbers that the domain can be composed of. Only real numbers, whole numbers, integers, even numbers, or odd numbers, are a few examples of typical domain restrictions. All of the numbers that fit or solve the x position is considered to be the domain of that function.
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Types
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There are three basic types of functional relationships; one-to-one, many-to-one, and one-to-many. In a one-to-one function, one number from the domain (input) results in only one number in the range (output). An example of a one-to-one function is the classic equation, y=2x+1.
In a many-to-one function, more than one number from the domain (input) results in only one number in the range (output). This is still a function, even though there are two or more domain numbers resulting in the same range number. An example of a many-to-one function is, y=x^2 or y=x^4.
In a one-to-many relationship, the result is not a function because one number from the domain (input) cannot result in more than one number in the range (output) and still be considered a function. An example of a one-to-many equation is, y^2=x or y^4=x.
Considerations
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The key element to making a function work is the content of the domain because it defines the limits of the function. The domain contains all the values for the desired range results. A function is a display of the relationship between two variables---the values of x as they relate to y.
Effects
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Once the values of a domain are selected or given, they are written as a set, with a capital letter, an equals sign, followed by the domain values enclosed in braces. For example, a domain may look like this: A={0, 1, 4, -4}. Plug these values into the functional equation to result in the "y" or range values.
It is standard practice in mathematics to write the domain values, along with the range values, to form ordered pairs, which is to match up the x number with its resulting y value. For example, if I take the domain above: A={0,1,4,-4}, and say it is the domain of the function: y=2x, once I plug in my x values, it will yield the following range:
B={0, 2, 8, -8}.
For every x value, I can match it with its corresponding y value to get the ordered pairs:
(0,0) (1,2) (4,8) (-4, -8).
Practice writing the ordered pair for domains and their corresponding ranges. Knowing how to write the ordered pairs is useful for the next step of learning how to plot these coordinates on a grid to see the resulting graph of the function.
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Resources
- Photo Credit Lake Tahoe Community College, Donna Roberts
