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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.

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 - x^{x} (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 - x^{x}, 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

= 2 - 1

= 1

and

f(1) = 2 - 1^{2}

= 2 - 1

= 1

= 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.

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

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

For y = 11:

11 = 3x^{2} - 1

11 + 1 = 3x^{2}

12 = 3x^{2}

x^{2} = *12**/**3*

x^{2} = 4

x = √4

x = -2 and x = 2

11 + 1 = 3x

12 = 3x

x

x

x = √4

x = -2 and x = 2

For y = 26:

26 = 3x^{2} - 1

26 + 1 = 3x^{2}

27 = 3x^{2}

x^{2} = *27**/**3*

x^{2} = 9

x = √9

x = -3 and x = 3

26 + 1 = 3x

27 = 3x

x

x

x = √9

x = -3 and x = 3

For y = 47:

47 = 3x^{2} - 1

47 + 1 = 3x^{2}

48 = 3x^{2}

x^{2} = *48**/**3*

x^{2} = 16

x = √16

x = -4 and x = 4

47 + 1 = 3x

48 = 3x

x

x

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.

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