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# Difference Between CoDomain and Range

July 17, 2022

Both CoDomain and Range are the terms associated with functions used in mathematics. While both are related to output and are often used interchangeably,  there is a difference between the two. In this article let’s understand the difference between codomain and range.

## What is Domain in Function?

The domain of a function is the set of values that we are allowed to plug into our function. This set is the $x$ values in a function such as $f(x)$.

## What is CoDomain in Function?

The codomain of a function is the set of possible values that the function can output. This set contains the values from which the values of range are taken.

## What is Range in Function?

The range of a function is the set of values that the function outputs. This set contains the values that the function gives out after we plug an $x$ value in.

In short,

• domain: what can go into a function
• codomain: what may possibly come out of a function
• range: what actually comes out of a function

If $f: A → B$, then

• Set $A$ is known as the domain of the function $f$
• Set $B$ is known as the co-domain of the function $f$
• A set of all $f$-images of all the elements of $A$ is known as the range of $f$. Thus, the range of $f$ is denoted by $f(A)$.

Let’s consider an example to understand these three terms.

Consider a function $f: N → N, f(x) = x^2$

• Domain $= N = \{1, 2, 3, 4, 5, 6, …\}$
• CoDomain $= N = \{1, 2, 3, 4, 5, 6, …\}$
• Range $= f(x) = \{1, 4, 9, …\}$

## Difference Between CoDomain and Range

Codomain is simply the set of values that includes a range along with a set of additional values. Understanding the different definitions is important as it helps to clarify the differences between one and the other.

Usually, Codomain and Range serve the same purpose when it comes to figuring out the output of the function. Using a Domain and Range calculator, one can easily find out the solutions to any problems presented to them.

## Conclusion

Although codomain and range both refer to the output of a function. But there is a difference between the two. The range is a set formed by the actual output of a function, whereas the codomain is a superset of a range.