Onto Functions in Mathematics

Onto Function is one of the many types of functions defined based on the relationship between its domain and codomain. For any function to be onto, it needs to relate two sets with a very specific mapping between elements, meaning that each element of the codomain has at least one element in the domain as its pre-image. In simple words, for any function, if all the elements of the codomain are mapped to some element of the domain, then the function is said to be an onto function. 

In this article, we will discuss the concept of onto or surjective function in detail including its definition, example, and many more. We will also discuss key differences between one one, onto and into functions as well.

What is an Onto Function?

An onto function, also known as a surjective function, is a special type of mathematical rule that connects two sets, let's call them Set A and Set B. In an onto function, every element in Set B has a partner in Set A. This means that the function covers all the elements in Set B and there are no leftovers in Set B without a matching element in Set A.

Onto Function Definition

function f: A →B to be onto, for every y in the codomain B, there exists an x in the domain A such that: f(x) = y

Representation for Onto Function

The following illustration provides the representation of an example of a onto function.

In the figure, you can see that every element in Set B connects with an element in Set A. There's no element in Set B that is left unmatched in Set A.

Examples of Onto Function

Some examples of Onto functions are:

• Linear Functions: f(x) = 2x, g(x) = 5x + 5, etc.
• Quadratic Functions: f(x) = x²
• Polynomial Functions: f(x) = x³ - x
• Exponential Functions: f(x) = e^x, g(x) = 2^x, etc.
• Absolute Value Function: f(x) = |x|
• Trigonometric Functions: f(x) = sin x, g(x) = cos x, etc.

Note: Onto Functions are codomain dependent, as the above functions are onto if codomain is R i.e., Real Numbers. 

Properties of Onto Function

There are various properties of onto functions,

• An onto function makes sure that in set B (codomain), we use every element.
  • For example, if set B has numbers {1, 2, 3}, an onto function from set A to set B will guarantee that every number in set B has at least one number in set A connected with it.
• Think about a function that takes numbers from set A and doubles them to get numbers in set B. (f(x) = 2x)
  • If B = {2, 4, 6}, we can find the right inverse by dividing each number in set B by 2 to figure out where it came from in set A: {2/2 = 1, 4/2 = 2, 6/2 = 3}. (f^-1(x) = x/2)
• Imagine a function that matches students in set A to their favourite subjects in set B.
  • If students only like {Math, Science, English}, we can describe the function by listing only these subjects, not all the possible subjects.

Composition of Onto Function

In terms of composition, if you have two functions f: A → B and g: B → C, and both functions are onto (surjective), then their composition (g∘f): A → C will also be onto. 

This is because the composition of two onto functions ensures that every element in the codomain of the second function has a pre-image in the domain of the first function, and then in the codomain of the second function.

Note: If either of the two functions in the composition is not onto, the composition may or may not be onto. 

Learn More, Composition of Functions

Onto Function Graph

As all the elements in the codomain of a onto function have a pre-image in the domain, the graph of the onto function covers the entire codomain without leaving any gaps. Let's see some graphs of onto functions in the following illustration:

Number of Onto Functions

Sometimes, we want to know how many functions we can create between two sets, A and B. Both sets have some elements in common. Set A has |A| element and Set B has |B| elements. We can find the number of Onto functions from A to B using this formula:

Number of onto functions = (Total number of functions) - (Number of functions that are not onto)

Here's how it works:

• The total number of functions from A to B amounts to |B|^|A| because each thing in A can be paired with anything in B.
• To find the number of functions that aren't onto, we use the inclusion-exclusion principle.

One to One and Onto Functions

Let's clear up the difference between one-to-one (injective) and onto (surjective) functions:

• One-to-One (Injective) Function: An injective function is more like a careful matchmaker. It makes sure no element in Set B has more than one corresponding element in Set A. It doesn't necessarily cover all of Set B.
• Onto (Surjective) Function: A surjective function is like a good host at a party. It makes sure everyone in Set B has a corresponding element in Set A. It covers the entire Set B.

You can read detailed differences between both functions in the article: One-to-One Functions

Onto and Into Function

There are various differences between both onto and into functions, these differences are listed in the following table:

Read More,

• Bijective Function
• Relation and Function
• Domain and Range of Function

Solved Examples on Onto Functions

Example 1: Consider the function f that turns numbers from {1, 2, 3} into numbers from {2, 4, 6} by doubling them, defined as f(x) = 2x. Is this an onto function?

Solution:

To prove that f(x) = 2x is an onto function, we need to show that for every number in {2, 4, 6}, there's a number in {1, 2, 3} that, when doubled, gives the number in question. For each number in the set {2, 4, 6}, we can easily find a number in {1, 2, 3} that works. For instance, if we take 4, we can choose 2 from {1, 2, 3} and when we double 2, we get 4. This holds true for all numbers in {2, 4, 6}, so the function is indeed onto because it covers the entire set {2, 4, 6}.

Example 2: Let's examine the function f(x) = 2x, where x can be any real number. Is this function onto?

Solution:

To show that f(x) = 2x is onto, we must demonstrate that for any real number y, there exists a real number x such that f(x) = y. In this case, for any real number y, selecting x = y/2 satisfies the condition. Therefore, the function is indeed onto because it covers all real numbers.

Example 3: Consider the function g(x) = x², where x can be any integer. Is this an onto function?

Solution:

No, the function g(x) = x² is not onto because it doesn't cover the entire range of integers in the codomain. For example, there is no integer x that results in g(x) = -1 because the square of any integer is non-negative. This means that some values in the codomain, like -1, are not covered by this function, so it's not onto.

Example 4: Imagine we have the numbers 1, 2 and 3 in Set A and the numbers 2, 4 and 6 in Set B. We have a magic rule that says we double the number in Set A to get a number in Set B. So, is this magic rule an onto function? (f(x) = 2x)

Solution:

Yes, it sure is! We can see in the figure that every number in Set B (2, 4 and 6) has corresponding number in Set A (1, 2 and 3). So, this is an onto function.

Example 5: Let's take Set A as a set of shapes (Circle, Square, Triangle) and Set B as a set of colors (Red, Blue, Yellow, Green). We have a rule that links each shape in Set A with a color in Set B, such that Circle maps to Red, Square maps to Blue and Triangle maps to Yellow. Is this rule an onto function?

Solution:

No, it is not an onto function. While all shapes in Set A (Circle, Square, Triangle) are associated with colors in Set B (Red, Blue, Yellow), the color Green (from Set B) has no corresponding shape in Set A. In an onto function, every element in the codomain (Set B) should have a pre-image in the domain (Set A), which is not the case here.

Example 6: Consider Set A as a set of animal names (Lion, Tiger, Bear) and Set B as a set of sounds (Roar, Growl). We have a rule that assigns animal names in Set A to the sounds they make in Set B, such that Lion maps to Roar, Tiger maps to Growl and Bear maps to Roar. Is this rule an onto function?

Solution:

Yes, it is! All the sounds in Set B (Roar, Growl) have corresponding animal names in Set A (Lion, Tiger, Bear). Therefore, this rule is an onto function.

Practice Problems on Onto Functions

Problem 1: Determine whether the function f: R → R defined by f(x) = 2x - 3 is onto.

Problem 2: Let g: Z → Z be defined by g(n) = 3n. Is the function g onto?

Problem 3: Consider the function h: {1, 2, 3, 4} → {5, 6, 7, 8} defined by h(1) = 5, h(2) = 6, h(3) = 7, and h(4) = 8. Is the function h onto?

Problem 4: Determine whether the function k: R → R defined by k(x) = x² is onto.

Problem 5: Let m: N → N be defined by m(n) = n + 1. Is the function m onto?

Problem 6: Consider the function p: {a, b, c, d, e} → {1, 2, 3} defined by p(a) = 1, p(b) = 1, p(c) = 2, p(d) = 2, and p(e) = 3. Is the function p onto?