# 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.

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.

## 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:

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.

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

Here is another example:

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