Many One Functions

Last Updated : 26 Dec, 2023

Many-to-One or Many One Function is one of the various types of functions that represent relationships between different entities. As we know, a function is a very specific type of relation in which each input has a unique output. In a many-one function, many inputs can have the same output. For example, x², where both 1 and -1 have the same output, i.e., 1.

In Simple terms, many to one function means When multiple elements are related to one element, or where multiple inputs can give the same output, that relation is known as Many to one function. Some examples of many-to-one functions are trigonometric functions like sine, cosine, and tangent, etc.

Many-one-Function

Table of Content

What is Many One Function in Maths?
Example of Many to One Function
Graph of Many to One Function
Properties of Many One function
One One and Many One Function

What is Many One Function in Maths?

Many to One function is one of the Types of function, also known as many one function, is where one or more elements of set A are related to only one element of Set B.

In other word, in any domain for any function, if two or more input is related to one single output, that relation is defined as many to one function.

For Example, If domain set A = {11, 12, 13, 14, 15 } and range set B = {p,q,r}. If this function is related as f = {(11,p) , (12,p) , (13,p) , (14,p) , (15,p) , (12,q) , (13,r) }. In this example, every number in domain set is related to p, which makes many to one function. If each member in Range (Set B) is used as output, it is also known as onto function.

Many-to-One Function Definition

Any function f: A → B is said to be many one function if there exist at least two non-distinct elements a₁and a₂in A such that f(a₁) = f(a₂).

In other words, if there are at least two different inputs in the domain that produce the same output in the codomain, then the function is many-to-one.

Read More about Function.

Example of Many to One Function

Some of the most common examples of many one funcitons are:

Absolute Value Function

Domain: All real numbers (ℝ)
Codomain: Non-negative real numbers ([0, ∞))
Function: f(x) = |x|

Square Function

Domain: All real numbers (ℝ)
Codomain: Non-negative real numbers ([0, ∞))
Function: f(x) = x²

Floor Function

Domain: All real numbers (ℝ)
Codomain: All Integers (ℤ)
Function: f(x) = ⌊x⌋

Floor Function

Domain: All real numbers (ℝ)
Codomain: All Integers (ℤ)
Function: f(x) = ⌈x⌉

Graph of Many to One Function

Graph of a many-to-one function doesn’t pass the horizontal line test for at least one point in its range. To check whether a function is many-one or not, we only have to draw a line parallel to the x-axis on the graph. If it intersects the graph at more than one point, then the function is a many-one function.

Let’s consider an example of a many-one function, i.e., f(x) = x². As x² maps both 1 and -1 to 1, it is an example of a many-to-one function. You can see the graph for this function below:

Graph of an Many One Function

Properties of Many One function

Some of the common properties of Many-One Function are listed as follows:

Many to one function cannot have inverse function.
The domain of the function should have at least two elements having the same codomain value.
There can be many inputs but only one output. Different input (Domain) can have the same output.
If every value in Range has input image, then many to one function is also called onto function.
The many one function can also be called a constant function if there is only one codomain.
The number of elements in the domain of many one functions should always be more than the number of elements in the codomain.

One One and Many One Function

The key differences between one-one and many-one function are listed in the following table:

One One Function Vs Many One Function
Feature	One-to-One Function (Injection)	Many-to-One Function
Definition	Each element in the domain maps to a unique element in the codomain.	At least two distinct elements in the domain map to the same element in the codomain.
Relationship between inputs and outputs	Precise one-to-one correspondence. No two inputs produce the same output.	Multiple inputs can produce the same output.
Horizontal Line Test	No horizontal line intersects the graph more than once.	A horizontal line can intersect the graph multiple times.
Example	f(x) = x, f(x) = x³, etc.	f(x) = x², f(x) = sin(x), etc.

Solved Examples of Many One Function

Example 1: If Domain = {1,2,3,4,52} and Codomain = {A, B, C} and they are mapped as f = {(1,A), (2,A), (3,A), (4,A), (52,B)}. Verify if they are Many to one function.

Solution:

As we can see, A is related to many elements in Domain. This is a classic Example of Many to one Function.

Example 2: Find if Parabola is a many to one function.

Solution:

General form of parabola is f(y) = ax² + bx + c

When we make a graph, we get two intersections with respect to x axis.

Hence parabola is many to one function.

Example 3: Show that the function f:R->R defined by f(x) = 5x² + 4, Prove that R is many to one function.

Solution:

lets check the value of -1 and 1 in the given equation.

f(-1) = 5× (-1)² + 4 = 9, and

f(1) = 5 × (1)²+ 4 = 9

They both have the same value for two different values. The different sets have an equal image. Hence, the equation is many to one function.

Example 4: In this example, we have a function f such that f(x) = x² + 9. Determine whether this function is many to one function.

Solution:

To check if the function is many to one or not, we will check the value of the equation and get some values of Domain and Range.

f(3) = 3 × 3 + 9 = 18,

f(-3) = (-3 × -3) + 9 =18,

f(4) = (4 × 4) + 9 = 25, and

f(-4) = (-4 × -4) + 9 = 25

Here we got the Set A = (3,-3, 4 , -4) Set B = (18, 25) and is an example of many to one function.

Practice Problems on Many to One Functions

Problem 1: If Domain = {13, 21, 31, 412, 2} and Codomain = {A, B, C} and they are mapped as f= {(13, A) , (21, A) , (31, A) , (412, A) , (2, B)}. Verify if they are Many to one function.

Problem 2: In this example, we have a function f such that f(x) = x² + 45x. Determine whether this function is many to one function.

Problem 3: In this example, we have a function f such that f(x) = x² + 19. Determine whether this function is many to one function.

Problem 4: Prove that Sine function is Many to One function?

Problem 5: Prove that Cosine function is Many to One function?

Problem 6: Explain Domain, Range, Codomain with visually?