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Home / Mathematics / One-to-One and Onto Functions

SAT One-to-One and Onto Functions in Mathematics

Last Updated on Mar 03, 2025
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One-to-One Functions

A one-to-one function is a type of function that maps each element in the range to only one element in its domain, which means that the outputs are unique and never repeated. 

For instance, the function is a one-to-one function because it provides a distinct output for every input. However, the function is not one-to-one because it generates the same output of for both and as inputs. If a function does not follow the one-to-one property, it is termed as a many-to-one function.

Let us visualize this by mapping two pairs of values to compare functions that are and that are not one to one.

In the Fig (a) (which is one to one), is the domain and is the codomain, likewise in Fig (b) (which is not one to one), is a domain and is a codomain.

  • In Fig(a), for each value, there is only one unique value of and thus, is one to one function. 
  • In Fig (b), different values of , , and are mapped with a common value and (also, the different values and are mapped to a common value ). Thus, is a function that is not a one to one function.

Onto Functions

If a function f maps from a set to a set , it is considered an onto function when every element has at least one corresponding element such that . An onto function ensures that no element is left unmapped in set , as they are all mapped to an element in set . The following example illustrates this concept:

Let and then .

What is One-to-One and Onto Function?

A function is said to be both one-to-one and onto if it satisfies both properties. In other words, each element in the domain is paired with a unique element in the range, and every element in the range is mapped to by at least one element in the domain. 

Graphically, a one-to-one and onto function is a function where every point in the range is covered by exactly one point in the domain, and no horizontal line intersects the graph more than once. 

For example, the function is a one-to-one and onto function because it maps every value of to a unique value of , and every value of is obtained by plugging in a value of .

How to Check One-to-One and Onto Function?

To check if a function is one-to-one and onto, follow these steps:

Checking for One-to-One:

Step 1: Assume that , where and are two different values in the domain of the function.

Step 2: Simplify the equation and see if it leads to a contradiction or forces .

Step 3: If the equation leads to a contradiction, the function is one-to-one.

Step 4: If the equation forces , the function is not one-to-one.

Checking for Onto:

Step 1: Determine the range of the function by evaluating for different values of in the domain.

Step 2: If every element in the range has a corresponding element in the domain, the function is onto.

Step 3: If there exist elements in the range that do not have a corresponding element in the domain, the function is not onto.

By following these steps, you can assess whether a function is one-to-one or onto.

One-to-One and Onto Function Graph

To provide a visual understanding of one-to-one and onto functions, here are the graphical representations of each:

One-to-One Function Graph:

The horizontal line test is used to determine whether a function is one-one when its graph is given. to test whether the function is one-one from its graph,

  • To determine if a function is one-to-one, imagine a horizontal line (like a stick) passing through the graph. 
  • If the line never intersects more than one point on the graph at any given moment, then the function is one-to-one. 
  • However, if the line does intersect multiple points of the graph at any instance, then the function is not one-to-one.

For example, Consider the graph below.

  • The quadratic function does not satisfy the one-to-one property as it fails the horizontal line test. The horizontal line intersects multiple points on its graph, indicating that it is not a one-to-one function.
  • The function , which is a cubic function, satisfies the one-to-one property because it passes the horizontal line test. The horizontal line intersects the graph at only one point at any given time, demonstrating that the function is one-to-one.

Onto Function Graph:

To determine if a function is onto using its graph, a straightforward approach is to compare the range with the codomain. If the range is equal to the codomain, then the function is onto. For any function graph, it can be considered onto only if every horizontal line intersects the graph at one or more points. If there exists an element in the range that fails to intersect the function's graph when subjected to the horizontal line test, then the function is not onto. The following image exemplifies a graph of an onto function:

Difference Between One-to-One and Onto Function

Here's a tabular format highlighting the key differences between one-to-one and onto functions:

Property

One-to-One Function

Onto Function

Definition

Each element in the domain maps to a unique element in the range.

Every element in the range has at least one corresponding element in the domain.

Mathematical notation

implies , for all , in the domain.

For every in the range, there exists an in the domain such that .

Inputs and Outputs

No two different inputs produce the same output.

No elements in the range are left unmapped; every element has a corresponding input.

Graphical representation

Passes the horizontal line test. The graph does not intersect any horizontal line at more than one point.

Covers the entire range vertically. The graph reaches every point in the range.

Injective or Surjective

Injective (one-to-one)

Surjective (onto)

Examples

Remember that a function can be both one-to-one and onto, which is referred to as a bijection.

One-to-One and Onto Function Solved Examples

1.Determine if the function is one-to-one or onto.

Solution:

To check if is one-to-one, assume that , where and are two different values in the domain of . Then we have:

Since and are equal, we can conclude that the function is one-to-one.

To check if is onto, we need to determine if every element in the range of has a corresponding element in the domain. The range of is all real numbers, since any real number can be obtained by plugging in a value for . 

Therefore, f(x) is onto.

2.Determine if the function is one-to-one or onto.

Solution:

To check if is one-to-one, we assume that , where and are two different values in the domain of . Then we have:

Since , either or . If , then , and we have proven that is one-to-one. However, if , then we cannot conclude that is one-to-one.

To check if is onto, we need to determine if every element in the range of has a corresponding element in the domain. The range of is all real numbers greater than or equal to . However, there is no real number that can be plugged in for to obtain a negative number in the range of . 

Therefore, is not onto.

3.Determine if the function is one-to-one or onto.

Solution:

To check if is one-to-one, assume that , where and are two different values in the domain of . Then we have:

Since and are equal, we can conclude that the function is one-to-one.

To check if is onto, we need to determine if every element in the range of has a corresponding element in the domain. The range of is all real numbers, since any real number can be obtained by plugging in a value for . 

Therefore, is onto.

Comprehending one-to-one and onto functions is critical in mastering higher-level mathematical principles and performing well on standardized tests like the SAT, ACT, and AP Calculus in the US. One-to-one and onto functions provide a solid foundation for the examination of intricate problems in algebra, calculus, and graph theory. With the determination of whether a function is injective, surjective, or bijective, students can be able to solve graph-based problems with confidence, establish domain and range, and test function transformations. A strong understanding of these principles gives students the problem-solving skills essential for both scholarly achievement and life beyond the classroom.

One-to-One and Onto Function FAQs

A one-to-one function maps each element of the domain to a unique element of the range, ensuring that no two different inputs produce the same output. On the other hand, an onto function maps every element in the range to at least one element in the domain, guaranteeing that no elements are left unmapped in the range.

Yes, a function can be both one-to-one and onto. In fact, such functions are referred to as bijections.

To determine if a function is one-to-one, you can use the horizontal line test. If no horizontal line intersects the graph of the function more than once, then the function is one-to-one. On the other hand, to verify if a function is onto, you need to check if every element in the range has a corresponding element in the domain such that the function maps to it.

One-to-one and onto functions have various applications in mathematics and other fields. One-to-one functions are useful in solving equations, cryptography, computer science, and modeling consumer preferences and demand. Onto functions find applications in physics, statistics, data analysis, and coding theory, among others.

Yes, it is possible for a function to be neither one-to-one nor onto. Such functions may have elements in the domain that map to the same element in the range (not one-to-one), and/or there may be elements in the range that are not mapped to by any element in the domain (not onto).

A function is said to be both one-to-one and onto if it satisfies both properties. In other words, each element in the domain is paired with a unique element in the range, and every element in the range is mapped to by at least one element in the domain
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