Many-to-one Function

Functions can be one-to-one or many-to-one relations.The many-to-one function states that the two or more different elements have the same image. Consider there are two sets A and B . If the elements of both these sets are enlisted, considering that the different elements of A have the same image in B, then it is known as the many-to-one function. 

The mathematical representation of the many-to-one function is as follows:

x1x2,f(x1)=f(x2)

If x1 and x2 are two different elements of a set and are not equal even if their F images are equal, they exhibit the many-to-one function.

Graphical representation of the many-to-one function is:

To check whether the graph is a one-to-one or many-to-one function, we will have to draw a line parallel to the X-axis. If it intersects with the graph at more than one point then it is known to have the many-to-one function. As you can see above, there are two intersecting points; hence it is the many-to-one function.
The many-to-one function is also called a constant function when all the elements of the set or domain are similar to or have the same image as only one element in the codomain.

Properties of the many-to-one function 

• The range value is always larger than the codomain.

• The codomain has the same value for more than one domain.

• The function’s domain should have at least two elements to have the codomain value.

• The number of elements in the codomains is less than the number of elements in the domain.

Solved examples of many-to-one functions

Examples1: f(x)=(x2) check if the function is many-to-one or not?

Solution: Domain = {1,-1,2,-2} 

Putting elements of domain in the function 

f(1)=12=1

f(-1)=(-12)=1

f(2)=(2)2=4

f(-2)=(-2)2=4

So, our codomain = {1,4}

If we map the domain and codomain, then we can see that one element of the domain has a single image. Hence this function is a many-to-one function.

Examples 2: A={1,2,3,4}, B={a,b,c,d,e}  function is defined as f={(1,a),(2,a),(3,b),(4,c)}. Check whether the function is many-to-one or not?

Solution: Representing the function through mapping

The condition for a function to be many-to-one, is that one or more than one element of the domain should have the same image in the codomain. As it is clear in the map above, the elements of domain {1,2} have the same image in the codomain {a}. Thus the function is a many-to-one function.

Example 3: f:XY={(1,x),(2,x),(3,x),(4,y),(5,z)} check whether the function is many-to-one or not.

Solution: As it is clear that the map above the elements of domain {1,2,3} have the same image in the codomain {x},  thus this function is many-to-one. 

Conclusion 

We learned about the many-to-one function and its properties in this article.

The many-to-one relation associates two or more input variable values with a single output variable value. You can also check whether a function is a one-to-one, or many-to-one function with the help of a graphical representation. First, draw a parallel line to the X-axis and if the line intersects at more than one point, it is a many-to-one function. Whereas, if the line intersects only at one point, the function is a one-to-one function.