# Math Lesson 16.2.3 - Surjective Function

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Welcome to our Math lesson on Surjective Function, this is the third 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.

## Surjective Function

In such functions, each element of the output set Y has at least one element of the input set X in correspondence. In other words, unlike in injective functions, in surjective functions, there are no free elements in the output set Y; all y-elements are related to at least one x-element.

For example, the function f(x) = 2 - xx (x ∊ R, y ∊ R) is surjective because for every y-value we can find its corresponding x-value. For example, f(2) = 2 means that the corresponding x-value is 2 = 2 - xx, so x = 0.

In surjective functions, we may have more than one x-value corresponding to the same y-value. For example, in the above function, we have two input values x = -1 and x = 1 that give the same output y because

f(-1) = 2 - (-1)2
= 2 - 1
= 1

and

f(1) = 2 - 12
= 2 - 1
= 1

The formal definition of surjective functions is as below:

A function f (from the input set X to the output set Y) is surjective only if for every y in Y, there is at least one x in X such that f(x) = y. In other words, the function f(x) is surjective only if f(X) = Y.

### Example 3

The output set Y of the surjective function y = 3xx - 1 is Y = {11, 26, 47}. Find all possible values of the input set X.

### Solution 3

Substituting the values of the output set Y in the formula of this function yields:

For y = 11:

11 = 3x2 - 1
11 + 1 = 3x2
12 = 3x2
x2 = 12/3
x2 = 4
x = √4
x = -2 and x = 2

For y = 26:

26 = 3x2 - 1
26 + 1 = 3x2
27 = 3x2
x2 = 27/3
x2 = 9
x = √9
x = -3 and x = 3

For y = 47:

47 = 3x2 - 1
47 + 1 = 3x2
48 = 3x2
x2 = 48/3
x2 = 16
x = √16
x = -4 and x = 4

Hence, we have the following set as input:

X = {-4, -3, -2, 2, 3, 4}

We can show this function through the Venn diagram method as follows:

You have reached the end of Math lesson 16.2.3 Surjective Function. 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.

## More Injective, Surjective and Bijective Functions. Graphs of Functions Lessons and Learning Resources

Functions Learning Material
Tutorial IDMath Tutorial TitleTutorialVideo
Tutorial
Revision
Notes
Revision
Questions
16.2Injective, Surjective and Bijective Functions. Graphs of Functions
Lesson IDMath Lesson TitleLessonVideo
Lesson
16.2.1Domain, Codomain and Range
16.2.2Injective Function
16.2.3Surjective Function
16.2.4Bijective Function
16.2.5The Graph of a Function
16.2.6Horizontal Line Test
16.2.7Function or not a Function? The Vertical Line Test

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