Codomain And Range

This article will give you a short note on codomain and range. Before we proceed, let’s quickly recap what a function is: A function relates each element of a set with exactly one element of another set(possibly the same set). That’s, a function relates an input to an output. To specify the types and values of a function input and output more clearly, we use the terms domain, codomain, and range.

Codomain And Range

The Codomain of a function is the set of its possible outputs. In the function machine metaphor, the codomain is the set of objects that might come out of the machine.

For example, when we use the function notation f: R – R, we mean that f is a function from the real numbers to the real numbers. In other words, the codomain of f is the set of real numbers R(and its set of possible inputs or domain is also the set of real numbers R).

Just because an object is in the codomain of a function, it does not necessarily mean that there is an input for which the function will output that object.For example,we could define a function f : R – R as f(x) = x2. Since f(x) will always be non-negative, the number -3 is in the codomain of f, but there is no input of x for which f(x) = -3.

The set of all outputs that result from putting all inputs into the function is called the range. For the above f, the range is the set of non-negative real numbers, while the codomain is the set of all real numbers.

Purpose of Codomain and Range

Co-domain of a function is a set of values that includes the range but may include some additional values. The purpose of the codomain is to restrict the output of a function. The range can be difficult to specify sometimes, but a larger set of values that include the entire range can be specified. The codomain of a function sometimes serves the same purpose as the range.

Examples of Codomain and Range

If A = { 1,2,3,4 } and B ={1,2,3,4,5,6,7,8,9 } and the relation f : A -> B is defined by f(x) = x^2,then codomain = Set B ={ 1,2,3,4,5,6,7,8,9 }

and Range = { 1,4,9 }.The range is the square of set A  but the square of 4 (that is 16) is not present in either set B (codomain) or the range.

Codomain vs Range: Comparison Chart

Summary of Codomain vs Range

Both terms are used in native set theory, but the difference between codomain and range is very subtle. Codomain is simply the set of possible output values for a function. It’s the result of a function, which is how mathematicians define it. It’s also possible to define a function’s range as the values that it returns. As a result, it is possible to use the term codomain incorrectly. But in modern mathematics, the range is described as a much broader subset of the codomain.

The importance of Codomain

Is the square root of a number a function?

The square root is not a function if the codomain (the possible outputs) is the set of real numbers. Are you surprised by that?

f(9) = 3 or -3 is an example of a situation where the same input could have two possible outcomes.

A single value can only be used for a function. If you enter the same input twice, you’ll get an error. “f(9) = 3 or -3” is incorrect!

However, limiting the codomain to non-negative real numbers is an easy fix.

Since the radical symbol (like x) always denotes the principal (positive) square root, x is a function by virtue of its correct codomain.

As a result, the codomain we choose can have an impact on whether or not something is considered a function.