Functions are used to model and analyze real world situations like future stock market tendencies or planetary motions. Among important properties of functions, knowing that a function is one-to-one and onto are vital, primarily for students attending competitive exams for the United States like the SAT, ACT, and AP Calculus. These concepts are the building blocks for solving advanced problems in algebra, calculus, and discrete mathematics. Understanding injective and surjective functions not only builds problem-solving ability but also prepares students to answer graph analysis, domain-range, and function transformation questions — areas commonly assessed in these exams. In this article, we will go deep into the topics of one-to-one and onto functions, learn about their importance, and identify their significance in learning high-level mathematics.

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 \(g(x) = x - 4\) is a one-to-one function because it provides a distinct output for every input. However, the function \(g(x) = x^{2}\) is not one-to-one because it generates the same output of \(4\) for both \(2\) and \(-2\) 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), \(x\) is the domain and \(f(x)\) is the codomain, likewise in Fig (b) (which is not one to one), \(x\) is a domain and \(g(x)\) is a codomain.

• In Fig(a), for each \(x\) value, there is only one unique value of \(f(x)\) and thus, \(f(x)\) is one to one function. 
• In Fig (b), different values of \(x\), \(2\), and \(-2\) are mapped with a common \(g(x)\) value \(4\) and (also, the different \(x\) values \(-4\) and \(4\) are mapped to a common value \(16\)). Thus, \(g(x)\) is a function that is not a one to one function.

Onto Functions

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

Let \(A = \left\{a_1, a_2, a_3\right\}\) and \(B = \left\{b_1, b_2\right\}\) then \(f: A\to B\).

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 \(f(x) = x\) is a one-to-one and onto function because it maps every value of \(x\) to a unique value of \(y\), and every value of \(y\) is obtained by plugging in a value of \(x\).

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 \(f(a) = f(b)\), where \(a\) and \(b\) 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 \(a = b\).

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

Step 4: If the equation forces \(a = b\), the function is not one-to-one.

Checking for Onto:

Step 1: Determine the range of the function by evaluating \(f(x)\) for different values of \(x\) 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 \(f(x) = x^{2}\) 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 \(f(x) = x^{3}\), 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:

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 \(f(x) = 2x - 3\) is one-to-one or onto.

Solution:

To check if \(f(x)\) is one-to-one, assume that \(f(a) = f(b)\), where \(a\) and \(b\) are two different values in the domain of \(f(x)\). Then we have:

\(2a - 3 = 2b - 3\)

\(2a = 2b\)

\(a = b\)

Since \(a\) and \(b\) are equal, we can conclude that the function \(f(x)\) is one-to-one.

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

Therefore, f(x) is onto.

2.Determine if the function \(g(x) = x^{2} - 4x\) is one-to-one or onto.

Solution:

To check if \(g(x)\) is one-to-one, we assume that \(g(a) = g(b)\), where \(a\) and \(b\) are two different values in the domain of \(g(x)\). Then we have:

\(a^{2} - 4a = b^{2} - 4b\)

\(a^{2} - b^{2} - 4a + 4b = 0\)

\((a - b)(a + b) - 4(a - b) = 0\)

\((a - b)(a + b - 4) = 0\)

Since \((a - b)(a + b - 4) = 0\), either \(a - b = 0\) or \(a + b - 4 = 0\). If \(a - b = 0\), then \(a = b\), and we have proven that \(g(x)\) is one-to-one. However, if \(a + b - 4 = 0\), then we cannot conclude that \(g(x)\) is one-to-one.

To check if \(g(x)\) is onto, we need to determine if every element in the range of \(g(x)\) has a corresponding element in the domain. The range of \(g(x)\) is all real numbers greater than or equal to \(-4\). However, there is no real number that can be plugged in for \(x\) to obtain a negative number in the range of \(g(x)\). 

Therefore, \(g(x)\) is not onto.

3.Determine if the function \(h(x) = 3x - 7\) is one-to-one or onto.

Solution:

To check if \(h(x)\) is one-to-one, assume that \(h(a) = h(b)\), where \(a\) and \(b\) are two different values in the domain of \(h(x)\). Then we have:

\(3a - 7 = 3b - 7\)

\(3a = 3b\)

\(a = b\)

Since \(a\) and \(b\) are equal, we can conclude that the function \(h(x)\) is one-to-one.

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

Therefore, \(h(x)\) 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.