One to One Function

One to one function basically denotes the mapping of two sets. With the help of examples or solving problems based on this concept, we are going to learn about this function, so that its concept could be easily understood. Apart from the one-to-one function, there are other sets of functions which denotes the relation between sets, elements or identities. They are;

  • Many to One function or Surjective function
  • Onto Function or Bijective function

Functions

A function f() is a method, which relates elements/values of one variable to the elements/values of another variable, in such a way that elements of the second variable is identically determined by the elements of the first variable. A function has many types and one of the most common functions used is the one-to-one function or injective function. Also, we will be learning here the inverse of this function.

One to One Function

Also, we have other types of functions in Maths which you can learn here easily, such as Identity function, Constant function, Polynomial function, etc. Let us now learn, a brief explanation with definition, its representation and example.

Definition of One-to-One Functions

One-to-One functions define that each element of one set say Set (A) is mapped with a unique element of another set, say, Set (B).

Or

It could be defined as each element of Set A has a unique element on Set B.

In brief, let us consider ‘f’ is a function whose domain is set A. The function is said to be injective if for all x and y in A,

Whenever f(x)=f(y), then x=y

And equivalently, if x ≠ y, then f(x) ≠ f(y)

Formally, it is stated as, if f(x)=f(y)  implies x=y, then f is one-to-one mapped or f is 1-1.

Similarly, if f is a function which is one to one, with domain A and range B, then the inverse of function f is given by;

f-1(y) = x ; if and only if f(x) = y

Horizontal Line Test: An injective function can be determined by the horizontal line test or geometric test.

  1. If a horizontal line can intersect the graph of the function, more than one time then the function is not mapped as one-to-one.
  2. If a horizontal line can intersect the graph of the function only a single time, then the function is mapped as one-to-one.

Example

Example: Let A = {1, 2, 3} and B = {a, b, c, d}. Which of the following is a one-to-one function?

  1. {1,a1,a, 2,c2,c, 3,a3,a}
  2. {1,b1,b, 2,d2,d, 3,a3,a}
  3. {1,a1,a, 2,a2,a, 3,a3,a}
  4. {1,c1,c, 2,b2,b, 1,a1,a, 3,d3,d}

The Answer is 2.

Explanation: Here, option number 2 satisfies the one-to-one condition, as elements of set B(range) is uniquely mapped with elements of set A(domain).

Students are advised to solve more of such example problems, to understand the concept of one-to-one mapping clearly. To learn more about functions topic, you can visit BYJU’S Maths section, where we have explained all types of functions. Also, download its app to get personalised learning videos.

 


Practise This Question

State whether true or false- Two equivalent fractions can never be like fractions.