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103 LearnersLast updated on October 17, 2025
Many One Functions
A many-one function is a function that maps two or more elements of the domain to the same element in the codomain. Many one function is written as f: x → y, such that two or more elements of the set x are related to a single element of the set y. We will learn more about one function in this article.
What is a Many-to-One Function?
A many-to-one function is a type of mathematical function where multiple inputs from the domain can have the same output in the codomain. In other words, a function is called many-to-one when two or more different inputs produce the same output. A function f: A → B is many-to-one if there exist at least two distinct elements, a1 and a2, in set A such that f (a1) = f (a2).
Example:
Set A = {1, 2, 3, 4} and set B = {x, y}
If f = {(1, x), (2, x), (3, x), (4, y)}
Here, 1, 2, and 3 give the same output as x. So, it is known as a many-to-one function.
Difference between One and Many One Functions
Functions can be classified into how the inputs, i.e., domain, are connected to the outputs, i.e., codomain. The table given below shows the difference between a one-to-one function and a many-to-one function.
|
Feature |
One-to-One Function |
Many-to-One Function |
|
Definition |
Every input is connected to a different output. No two inputs have the same result. |
Two or more different inputs give the same output. |
|
Inputs and Outputs |
Each input matches only one unique output. |
Different inputs can be connected to one common output. |
|
Horizontal Line Test |
A horizontal line crosses the graph only once. |
A horizontal line can cross the graph more than once. |
|
Example |
f(x) = x, f(x) = x3 |
f(x) = x2, f(x) = sin(x) |
Properties Of Many One Function
A many-to-one function is a type of function where more than one input gives the same output. These functions have some special properties that make them easy to identify. The important properties of many one functions are:
At least two different inputs from the domain have the same output in the codomain of this function.
The number of elements in the domain is greater than the number of elements in the codomain.
A single value from the codomain can be the result of more than one input from the domain.
The number of distinct outputs (the range) is less than or equal to the number of inputs (the domain), but the codomain can be larger than the range.
A function is called a constant function if all inputs from the domain produce the same single output value.
Graph of Many to One Function
Follow the steps below to see if a graph is one-to-one or many-to-one:
Step 1: Draw a horizontal line (parallel to the x-axis) across the graph.
Step 2: If the line touches the graph in more than one spot, that means different inputs are giving the same output — so it’s a many-to-one function.
Looking at the graph above, the horizontal line crosses the curve at two points, which means the function is many-to-one.
Real Life Applications of Many One Function
Many one functions are commonly seen in our daily lives, as many real-world situations involve grouping different inputs into a single result. They are used in various fields like education, business, science, and technology. Here are some of them explained below.
- Education: In school, many students are assigned to the same class teacher. Each student is considered an input, and the class teacher they are assigned to is the output. This mapping of several students to one teacher is an example of a many-to-one function.
- Products and Categories: In a supermarket, different products like bread, biscuits, and cakes all belong to the same category—bakery. Here, each product is the input, and the category it belongs to is the output. Since different products can belong to the same category, this is a many-to-one function.
- Banking: In banks, many account holders are linked to the same branch. For example, several customers may have accounts in the same “Main City Branch”. Thus, customers are mapped to one branch.
- Traffic Light Control: At a traffic signal or an intersection, different sensor inputs like time and vehicle counts map to one signal state (red, yellow, or green).
- Mail delivery: Many addresses within a geographic area map to a single postal or ZIP code for mail delivery.
Common Mistakes and How to Avoid Them in Many One Functions
When studying or solving problems related to many one functions, students often make simple errors due to confusion about the domain, co-domain, or mapping rules. Given below are some common mistakes and ways to avoid them:
Mistake 1
Confusing many-to-one and one-to-one
Assuming that each input must have a unique output. Students forget that in a many-to-one function, different inputs can share the same output. Always remember that in a many-to-one function, two or more inputs can have the same value as output. For example, A = {1, 2, 3} and B = {x, y}, f = {(1, x), (2, x), (3, y)} is a many-to-one function.
Mistake 2
Believing that all elements of the codomain must be used
Students think that all the elements from the codomain set must be used, which is wrong. A function does not require every element of the codomain to be mapped. If A = {1, 2} and B = {x, y, z}, f = {(1, x), (2, x)} is valid even if y and z are not used.
Mistake 3
Confusing domain and codomain
Students sometimes mix up which set is the domain, which is the codomain. Remember that the domain is the input set and the codomain is the possible outputs. In A = {a, b} and B = {x, y}, A is the domain set.
Mistake 4
Using the same input for two different outputs
Using the same input for two different outputs breaks the definition of a many-to-one function. Many-to-one functions cannot have one input for two outputs. We can avoid this mistake by checking each input in the domain maps to exactly one output. For example, f = {(1, x), (1, y)} is wrong.
Mistake 5
Thinking constant functions are never many-to-one
Thinking that the constant function is not a many-to-one function is not correct. A constant function is a special case of a many-to-one function. A constant function occurs when all inputs give the same output. Avoid this confusion by checking if all the inputs map to one value. If f = {(1, x), (2, x), (3, x)} is both constant and a many-to-one function.
Solved Examples of Many One Functions
Problem 1
Let A = {1, 2, 3, 4} and B = {x , y}. Define f: A → B as f = {(1, x), (2, x), (3, y), (4, y)}. Is f a many-to-one function?
Yes, f is a many-to-one function
Explanation
Here, the inputs 1 and 2 have the result as x, and inputs 3 and 4 both map to y. Since at least two elements in A have the same output in B, this is a many-to-one function.
Problem 2
For the function f(x) = x2, where x ∈ {-2, 0, 1, 2}, is the function many-to-one?
Yes, f(x) = x2 is a many-to-one function
Explanation
Evaluating,
f(-2) = 4
f(-1) = 1
f(0) = 0
f(1) = 1
f(2) = 4
Here, f(-2) = f(2) and f(-1) = f(1). Since different inputs give the same outputs, the function is many-to-one.
Problem 3
Let sets be: A = {cat, dog, bird, rabbit} and B = {mammal, bird}. A function is defined as: f = {(cat, mammal), (dog, mammal), (rabbit, mammal), (bird, bird)}. Is this a many-to-one function?
Yes, it is a many-to-one function.
Explanation
The inputs cat, dog, and rabbit all map to the same output: mammal. Since different inputs give the same output, this is a many-to-one function.
Problem 4
Check if the function f(x) = x2 is many-to-one, for x ∈ {-2, -1, 0, 1, 2}.
Yes, f(x) = x2 is a many-to-one function.
Explanation
Evaluating:
f(-2) = 4
f(-1) = 1
f(0) = 0
f(1) = 1
f(2) = 4
Here, different inputs like -2 and 2 both give 4. Also, -1 and 1 both give 1.
This means multiple inputs share outputs, so it’s a many-to-one function.
Problem 5
Let sets be: A = {X, Y, Z} and B = {Red}. Function is defined as f = {(X, Red), (Y, Red), (Z, Red)}. What type of function is this?
It is a many-to-one function.
Explanation
All inputs give only one output (Red). That makes it a constant function, which is a special type of many-to-one function.
FAQs on Many One Functions
1.What is a many-to-one function?
2.How is a many-to-one function different from a one-to-one function?
3.Can a many-to-one function have repeated outputs?
4. Is every constant function a many-to-one function?
5.How do you identify a many-to-one function from a graph?
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Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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: He loves to play the quiz with kids through algebra to make kids love it.
