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.

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 a₁ and a₂ in 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.

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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:

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:

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?

Many One Functions: FAQs

1. What Do You Mean by Many-One Relation?

A many-one relation, also known as a many-to-one relation, occurs when multiple elements in the domain map to the same element in the codomain.

2. What Are the 4 Types of Functions?

Four types of functions are:

• One One or Injective Function
• Onto or Surjective Function
• One One and Onto or Bijective Function
• Into Function

Other than this there is one more type i, Many One Function.

3. What Is the Test for Many-One Function?

We can use horizontal line test to check wheather any function is many one function or not.

4. What Is the Difference Between Onto and Many-One Function?

• Onto Function: Every element in the codomain has at least one preimage in the domain.
• Many-One Function: Multiple elements in the domain can map to the same element in the codomain, and it may not cover the entire codomain.

5. What Is a Many-One Function Also Known As?

A many-one function is also known as a “multivalued function” or a “multivalued mapping.”

6. What are the Real-Life Examples of Many to One Function?

There are many real-life examples of many to one function like a student teacher relationship, many films one actor, many courses one student.