Math Lesson 16.2.1 - Domain, Codomain and Range

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Welcome to our Math lesson on Domain, Codomain and Range, this is the first lesson of our suite of math lessons covering the topic of Injective, Surjective and Bijective Functions. Graphs of Functions, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Domain, Codomain and Range

In the previous tutorial, we explained the meaning of domain and range. Thus, we said that domain D is the set of all values the independent variable (input) of a function takes, while range R is the set of the output values resulting from the operations made with input values.

However, there is another concept encountered very often when dealing with functions. It is called Codomain Y. It includes all possible values the output set contains. Given this description, it is clear that Range is a subset of (is included in) the Codomain. Look at the figure below.

The domain D determined by the input values is D = {1, 2, 3, 4} and range R of this function is R = {a, b, c, d}. However, the codomain contains an additional value 'e'. Hence, it is Y = {a, b, c, d, e}.

Example 1

Identify the domain, codomain and range in the functions given below if X is the input set and Y the output one.

1. X = {2, 4, 6, 8}, Y = {5, 9, 13, 17, 21} and f(x) = 2x + 1
2. X = {1, 2, 3, 4}, Y = {1, 4, 9, 16, 25, 36} and f(x) = x^x

Solution 1

1. From theory, it is known that domain D is the set of the input values of a function. In the specific case, it is represented by the X-set. Hence, the domain is D = X = {2, 4, 6, 8}.
   The range R is the set of the corresponding outputs obtained by making operations with the domain values. In the specific case, we have
   f(2) = 2 ∙ 2 + 1
   = 4 + 1
   = 5
   f(4) = 2 ∙ 4 + 1
   = 8 + 1
   = 9
   f(6) = 2 ∙ 6 + 1
   = 12 + 1
   = 13
   f(8) = 2 ∙ 8 + 1
   = 16 + 1
   = 17
   Once the corresponding outputs are identified, we can write the range E of this function. It is E = {5, 9, 13, 17}.
   Last, we have to find the codomain of this function. It includes all values contained in the output set. Hence, the codomain is Y = {5, 9, 13, 17, 21}.
2. Using the same reasoning as in (a), we obtain the domain D = X = {1, 2, 3, 4}.
   The corresponding output values for these inputs are
   f(1) = 1² = 1
   f(2) = 2² = 4
   f(3) = 3² = 9
   f(4) = 4² = 16
   Therefore, the range is R = {1, 4, 9, 16}.
   Last, we have to find the codomain of this function. It includes all values contained in the output set. Hence, the codomain is Y = {1, 4, 9, 16, 25}.

You have reached the end of Math lesson 16.2.1 Domain, Codomain and Range. There are 7 lessons in this physics tutorial covering Injective, Surjective and Bijective Functions. Graphs of Functions, you can access all the lessons from this tutorial below.

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