Domain in Math: Definition & Overview

Instructor: Jennifer Beddoe
The domain is the group of numbers that can be entered into a function to create a valid output. This lesson will further describe the domain and methods for determining the domain. A quiz at the end of a lesson will test your knowledge.

What is a Function?

A function describes the relationship between an input and an output. It can be compared to a machine that spits out different items depending on what was put into the machine. Let's take, for example, a fictional shoe machine. The operator puts in a colored disc, and, depending on the color of that disc, a certain shoe is created. So, if a white disc is placed in the machine, it creates a tennis shoe; a black disc creates a loafer; a yellow disc creates a sandal. It's the same each time; never does the machine create a tennis shoe from a yellow disc.

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In mathematics, functions are written as equations. They can also be shown as a graph.

Domain and Range

A function's input is called the domain, and a function's output is called the range. The domain relates to the colored discs from the example above, and the range would be the shoes that are created.

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Domain

The domain of a function is the set of all possible values that x can be equal to that will make a valid equation. There are only two instances in which an equation will not be valid - if there is a zero in the denominator or a negative square root. In all other instances, the equation works. Take, for example, the function f(x) = x^2.

No matter what value we substitute for x, the equation will be valid. Therefore, we would say that the domain of this function is all real numbers.

Determining the Domain

There are two methods used to determine the domain of a function: algebra and graphing.

Algebraically

Look at the following function:

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To find the domain of this function, we need to remember the definition of a domain and then do some simple algebra.

One thing we remember about the domain of a function is that it cannot include a negative square root. In order to find the domain, we have to know what numbers make the (x + 9) negative and exclude them. To do that, solve the equation:

x + 9 ≥ 0

x ≥ -9

The domain of this function is x ≥ -9.

Graphically

When you graph a function, the x-values signify the domain of that function. You can often determine the domain by looking at a graph.

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You can see from this graph that the domain of the function is all real numbers.

Here is another example:

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