A surjective function is any relation where each element belonging to the codomain is an image of at least one element of the domain. Also known as an envelope function, they are part of the classification of functions with respect to the way their elements are related.

For example a function F: A → B defined by F (x) = 2x

Which is read " F that goes from A to B defined by F (x) = 2x"

You have to define the starting and finishing sets A and B.

A: {1, 2, 3, 4, 5} Now the values ​​or images that each of these elements will yield when evaluated in F will be the elements of the codomain.

F (1) = 2

F (2) = 4

F (3) = 6

F (4) = 8

F (5) = 10

Thus forming the set B: {2, 4, 6, 8, 10}

It can be concluded then that:

F: {1, 2, 3, 4, 5} → {2, 4, 6, 8, 10} defined by F (x) = 2x It is a surjective function

Each element of the codomain must result from at least one operation of the independent variable through the function in question. There is no limitation of images, an element of the codomain can be an image of more than one element of the domain and still try a surjective function.

In the image 2 examples with surjective functions are shown.

Source: Author

In the first, it is observed that the images can be referred to the same element, without compromising the surjectivity of the function.

In the second we see an equitable distribution between domain and images. This gives rise to the bijective function, where the criteria of injective function and surjective function must be met .

Another method to identify surjective functions is to verify if the codomain is equal to the rank of the function. This means that if the arrival set is equal to the images provided by the function when evaluating the independent variable, the function is surjective.

Properties

To consider a function surjective, the following must be fulfilled:

Let F: D _f → C _f

∀ b ℮ C _f E a ℮ D _f / F (a) = b

This is the algebraic way to establish that for every “b” that belongs to C _f there is an “a” that belongs to D _f such that the function F evaluated in “a” is equal to “b”.

Surjectivity is a peculiarity of functions, where the codomain and range are similar. Thus, the elements evaluated in the function make up the arrival set.

Function conditioning

Sometimes a function that is not surjective can be subjected to certain conditions. These new conditions can make it a surjective function.

All kinds of modifications to the domain and codomain of the function are valid, where the objective is to fulfill the surjectivity properties in the corresponding relationship.

Examples: solved exercises

To meet the conditions of surjectivity, different conditioning techniques must be applied, this in order to ensure that each element of the codomain is within the set of images of the function.

Exercise 1

• Let the function F: R → R be defined by the line F (x) = 8 - x

A:

Source: author

In this case, the function describes a continuous line, which includes all real numbers in both its domain and range. Since the range of the function R _f is equal to the codomain R it can be concluded that:

F: R → R defined by the line F (x) = 8 - x is a surjective function.

This applies to all linear functions (Functions whose highest degree of the variable is one).

Exercise 2

• Study the function F: R → R defined by F (x) = x ²: Define if it is a surjective function. If not, show the conditions necessary to make it surjective.

Source: author

The first thing to take into account is the codomain of F, which is made up of the real numbers R. There is no way for the function to yield negative values, which excludes negative reals from the possible images.

Conditioning the codomain to the interval. It is avoided to leave elements of the codomain unrelated through F.

The images are repeated for pairs of elements of the independent variable, such as x = 1 and x = - 1. But this only affects the injectivity of the function, not being a problem for this study.

In this way it can be concluded that:

F: R → . This interval must condition the codomain to achieve the surjectivity of the function.

F: R → defined by F (x) = Sen (x) It is a surjective function

F: R → defined by F (x) = Cos (x) It is a surjective function

Exercise 4

• Study the function

F:).push ({});

Source: Author

The function F (x) = ± √x has the particularity that it defines 2 dependent variables at each value of "x". That is, the range receives 2 elements for each one that is made in the domain. A positive and negative value must be verified for each value of "x".

When observing the starting set, it is noted that the domain has already been restricted, this in order to avoid the indeterminacies produced when evaluating a negative number within an even root.

When checking the range of the function, it is noted that each value of the codomain belongs to the range.

In this way it can be concluded that:

F: [0, ∞) → R defined by F (x) = ± √x It is a surjective function

Exercise 4

• Study the function F (x) = Ln x denote if it is a surjective function. Condition the arrival and departure sets to fit the function to the surjectivity criteria.

Source: Author

As shown in the graph, the function F (x) = Ln x is defined for values ​​of "x" greater than zero. While the values ​​of "and" or the images can take any real value.

In this way we can restrict the domain of F (x) = to the interval (0, ∞)

As long as the range of the function can be kept as the set of real numbers R.

Considering this, it can be concluded that:

F: [0, ∞) → R defined by F (x) = Ln x It is a surjective function

Exercise 5

• Study the absolute value function F (x) = - x - and designate the arrival and departure sets that meet the surjectivity criteria.

Source: Author

The domain of the function is fulfilled for all real numbers R. In this way the only conditioning must be carried out in the codomain, taking into account that the absolute value function only takes positive values.

We proceed to establish the codomain of the function equal to the rank of the same

[0, ∞)

Now it can be concluded that:

F: [0, ∞) → R defined by F (x) = - x - It is a surjective function

Proposed exercises

1. Check if the following functions are surjective:

• F: (0, ∞) → R defined by F (x) = Log (x + 1)
• F: R → R defined by F (x) = x ³
• F: R → [1, ∞) defined by F (x) = x ² + 1
• [0, ∞) → R defined by F (x) = Log (2x + 3)
• F: R → R defined by F (x) = Sec x
• F: R - {0} → R defined by F (x) = 1 / x

References

1. Introduction to Logic and Critical Thinking. Merrilee H. Salmon. University of Pittsburgh
2. Problems in Mathematical Analysis. Piotr Biler, Alfred Witkowski. University of Wroclaw. Poland.
3. Elements of Abstract Analysis. Mícheál O'Searcoid PhD. Department of mathematics. University college Dublin, Beldfield, Dublind 4
4. Introduction to Logic and to the Methodology of the Deductive Sciences. Alfred Tarski, New York Oxford. Oxford University press.
5. Principles of mathematical analysis. Enrique Linés Escardó. Editorial Reverté S. A 1991. Barcelona Spain.